On some invariants of orbits in the flag variety under a symmetric subgroup

Abstract.

Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to $2$, let ${\mathcal{B}}$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in ${\mathcal{B}}$, we associate to every $K$-orbit on ${\mathcal{B}}$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\bf k}={\mathbb{C}}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in ${\mathcal{B}}$. We also characterize the invariants in terms of admissible paths in the set of $K$-orbits in ${\mathcal{B}}$.

1. Introduction

Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to $2$, and let ${\mathcal{B}}$ be the variety of Borel subgroups of $G$ with the conjugation action by $G$. Let $\theta$ be an order $2$ automorphism of $G$. The $\theta$-fixed point subgroup $K=G^{\theta}$ acts on ${\mathcal{B}}$ with finitely many orbits [12, 13, 16].

Let $V$ be the finite set of $K$-orbits in ${\mathcal{B}}$, and for $v\in V$, let $K(v)\subset{\mathcal{B}}$ be the corresponding $K$-orbit in ${\mathcal{B}}$. For a subset $X$ of ${\mathcal{B}}$, let $\overline{X}$ be the Zariski closure of $X$ in ${\mathcal{B}}$. The Bruhat order on $V$, denoted by $\leq$, is defined by

(1.1) $$v_{1}\leq v_{2}\hskip 14.454pt\mbox{if}\hskip 14.454ptK(v_{1})\subset\overline{K(v_{2})},\hskip 14.454ptv_{1},v_{2}\in V.$$

When ${\bf k}={\mathbb{C}}$, the geometry of the $K$-orbits and their closures in ${\mathcal{B}}$ plays an important role in the representation theory of real forms of $G$ via the Beilinson-Bernstein correspondence (see [9]). The poset $(V,\leq)$ and its application to representation theory have been the focus of extensive studies (see [1, 2, 3, 12, 13, 16, 17, 20]).

Let $W$ be the canonical Weyl group of $G$ with the set $S$ of canonical generators, and let $M(W,S)$ be the corresponding monoid (see $\lx@sectionsign$3.3 and $\lx@sectionsign$3.6). Among the structures on $V$ are the monoidal action of $M(W,S)$ on $V$ (see $\lx@sectionsign$3.6), the cross action of $W$ on $V$ (see $\lx@sectionsign$6.5), and the map $\phi:V\to W$ defined by Springer (see $\lx@sectionsign$6.6). The programs from the Atlas of Lie groups (www.liegroups.org) facilitate explicit computations in examples of the cross action, the monoidal action, and the Springer map.

Let $V_{0}=\{v\in V:K(v)\subset{\mathcal{B}}\;\mbox{is closed}\}$. In this paper, we associate to every $v_{0}\in V_{0}$ and $v\in V$ three subsets $Z_{v_{0}}(v)\subset Y_{v_{0}}(v)\subset W_{v_{0}}(v)$ of the Weyl group $W$. The set $W_{v_{0}}(v)$ is defined both geometrically in terms of intersections of $K(v)$ and Schubert varieties in ${\mathcal{B}}$ and combinatorially in terms of the Bruhat order on $V$ and the monoidal action of $M(W,S)$ on $V$, and $Y_{v_{0}}(v)$ (resp. $Z_{v_{0}}(v)$) is defined to be the set of minimal (resp. minimal length) elements in $W_{v_{0}}(v)$.

One of our main results (see Theorem 2.2 and Corollary 2.3) says that for any $v_{0}\in V_{0}$, the subset $Y_{v_{0}}(v)$ (or $Z_{v_{0}}(v)$) of $W$, together with the Springer invariant $\phi(v)\in W$, completely determine $v\in V$. This result generalizes [13, Theorem 5.2.2] of Richardson-Springer in the case when $(G,\theta)$ is of Hermitian symmetric type (see $\lx@sectionsign$9.3). When $V_{0}$ contains a single element $v_{0}$, the set $Y_{v_{0}}(v)$ for any $v\in V$ is a special case of an invariant for $K(v)$ studied by Springer in [17].

When $k={\mathbb{C}}$ and $K$ is the complexification of a maximal compact subgroup of a real form $G_{0}$ of $G$, we use the Matsuki duality between $K$-orbits and $G_{0}$-orbits in ${\mathcal{B}}$ to determine when a $G_{0}$-orbit and a Bruhat cell in ${\mathcal{B}}$ have non-empty intersection. The problem of determining when two such orbits intersect comes from Poisson geometry and served as the main motivation for this paper. See $\lx@sectionsign$2.3 and $\lx@sectionsign$5.

We introduce the notion of admissible paths from $v_{0}\in V_{0}$ to $v\in V$ by allowing only certain types of cross and monoidal actions by simple generators of $W$. We then characterize elements in $Y_{v_{0}}(v)$ (resp. $Z_{v_{0}}(v)$) as products of the simple generators in minimal (resp. shortest) admissible paths from $v_{0}$ to $v$. See $\lx@sectionsign$2.5 and $\lx@sectionsign$8.

The technical part of the paper is an analysis in $\lx@sectionsign$7 on the sets $Y_{v_{0}}(v)$ in relation to simple roots and their types relative to $v$. Various examples are computed in $\lx@sectionsign$9.

Precise statements of the main results in the paper are given in $\lx@sectionsign$2.

Most of the structures on $V$ are defined in [12, 13, 16] using a standard pair, i.e., a pair $(B,H)$, where $B\in{\mathcal{B}}$ and $H\subset B$ a maximal torus of $G$ such that $\theta(B)=B$ and $\theta(H)=H$. However, to formulate the sets $Z_{v_{0}}(v)\subset Y_{v_{0}}(v)\subset W_{v_{0}}(v)$, it is crucial that these structures can be defined independently of the standard pairs. Following [13, $\lx@sectionsign$1.7], we review in $\lx@sectionsign$3 and $\lx@sectionsign$6 the canonical definitions of all the structures on $V$ needed in the paper.

Acknowledgments

We would like to thank Xuhua He and George McNinch for helpful discussions. We thank the organizers of the Atlas of Lie Groups meeting in Utah in July 2010, which helped the development of this paper, and also thank Jeff Adams, Peter Trapa, and David Vogan for very useful discussions. We are especially grateful to Scott Crofts, who wrote a program allowing us to compute the invariants $Z_{v_{0}}(v)$ and $Y_{v_{0}}(v)$ explicitly in examples. The work described in this paper by the first author was partially supported by NSA grants H98230-08-0023 and H98230-11-1-0151, and that by the second author by the RGC grants HKU 7034/05P and 7037/07P of the Hong Kong SAR, China.

2. Statements of Results

2.1. Geometric and combinatorial definitions of $W_{v_{0}}(v)\subset W$

For $v_{0}\in V_{0}$ and $v\in V$, we define $W_{v_{0}}(v)\subset W$ by

(2.1) $$W_{v_{0}}(v)=\{w\in W:K(v)\cap\overline{B(w)}\neq\emptyset\},$$

where $B$ is any Borel subgroup of $G$ contained in $K(v_{0})$, and for $w\in W$, $B(w)$ is the corresponding $B$-orbit in ${\mathcal{B}}$ (see $\lx@sectionsign$3.5). The set $W_{v_{0}}(v)$ depends only on $v_{0}$ and $v$ and not on the choice of $B\in K(v_{0})$ (see Lemma 4.1).

Denote the monoidal action of $M(W,S)=\{m(w):w\in W\}$ on $V$ by $m(w)\!\cdot\!v$ for $w\in W$ and $v\in V$ (see $\lx@sectionsign$3.6). Our Lemma 4.2 gives the following combinatorial description of the set $W_{v_{0}}(v)$ for $v_{0}\in V$ and $v\in V$:

(2.2) $$W_{v_{0}}(v)=\{w\in W:\;v\leq m(w)\!\cdot\!v_{0}\}\subset W,$$

where recall that $\leq$ is the Bruhat order on $V$ defined in (1.1).

2.2. The subsets $Z_{v_{0}}(v)\subset Y_{v_{0}}(v)$ of $W_{v_{0}}(v)$

Let $l:W\to{\mathbb{N}}$ and $\leq$ be respectively the length function and the Bruhat order on $W$ as a Coxeter group (see $\lx@sectionsign$3.3 and $\lx@sectionsign$3.4). Let $W_{1}$ be a subset of $W$. An element $w\in W_{1}$ is said to be minimal if for any $w_{1}\in W_{1}$, $w_{1}\leq w$ implies that $w_{1}=w$. The set of all minimal elements in $W_{1}$ will be denoted by $\min(W_{1})$. An element $w\in W_{1}$ is said to have minimal length if $l(w)\leq l(w_{1})$ for every $w_{1}\in W_{1}$. The set of all minimal length elements in $W_{1}$ is denoted by $\min_{l}(W_{1})$.

For $v_{0}\in V_{0}$ and $v\in V$, define

(2.3) $$Y_{v_{0}}(v) =\min(W_{v_{0}}(v))\subset W_{v_{0}}(v),$$

(2.4) $$Z_{v_{0}}(v) ={\rm min}_{l}(W_{v_{0}}(v))={\rm min}_{l}(Y_{v_{0}}(v))\subset Y_{v_{0}}(v).$$

We prove in Lemma 4.3 that for any $v_{0}\in V$ and $v\in V$, $W_{v_{0}}(v)$ is determined by its subset $Y_{v_{0}}(v)$ in the sense that for any $w\in W$,

(2.5) $$w\in W_{v_{0}}(v)\hskip 14.454pt\mbox{iff}\hskip 14.454pty\leq w\;\;\mbox{for some}\;y\in Y_{v_{0}}(v).$$

Thus both $Z_{v_{0}}(v)$ and $W_{v_{0}}(v)$ are determined by $Y_{v_{0}}(v)$. The example in $\lx@sectionsign$9.4 shows that $Z_{v_{0}}(v)$ may be a proper subset of $Y_{v_{0}}(v)$.

Using the programs available at the Atlas of Lie groups, Scott Crofts has written a program that allows one to compute the sets $Y_{v_{0}}(v)$ explicitly in examples.

2.3. Intersections of real group orbits and Bruhat cells in ${\mathcal{B}}$

Assume now that ${\bf k}={\mathbb{C}}$ and that $K$ is the complexification of a maximal compact subgroup of a real form $G_{0}$ of $G$. For $v\in V$, let $G_{0}(v)\subset{\mathcal{B}}$ be the $G_{0}$-orbit in ${\mathcal{B}}$ that is dual to $K(v)\subset{\mathcal{B}}$ under the Matsuki duality [10] between $G_{0}$-orbits and $K$-orbits in ${\mathcal{B}}$. The following Theorem 2.1 is the first main result of this paper.

Theorem 2.1.

Let $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. Then for $v\in V$ and $w\in W$,

$$G_{0}(v)\cap B(w)\neq\emptyset\hskip 14.454pt\mbox{iff}\hskip 14.454ptw\in W_{v_{0}}(v).$$

Our motivation for Theorem 2.1 comes from Poisson geometry: when $G_{0}$ is connected, it is shown in [7] that there is a Poisson structure $\Pi$ on ${\mathcal{B}}$ such that the connected components of intersections of $G_{0}$-orbits and $B$-orbits in ${\mathcal{B}}$ are precisely the $H_{0}$-orbits of symplectic leaves of $\Pi$, where $H_{0}=B\cap G_{0}$ is a maximally compact Cartan subgroup of $G_{0}$. Thus, one first needs to know when the intersection of a $G_{0}$-orbit and a $B$-orbit in ${\mathcal{B}}$ is non-empty. In view of (2.5), Theorem 2.1 gives a complete answer to this question in terms of the set $Y_{v_{0}}(v)$ and the Bruhat order on $W$. Further applications to the Poisson structure $\Pi$ on ${\mathcal{B}}$ will appear elsewhere.

2.4. $Y_{v_{0}}(v)$ and $Z_{v_{0}}(v)$ as invariants of $K(v)$

Our second main result, the following Theorem 2.2, implies that the pair $(\phi(v),Y_{v_{0}}(v))$ forms a complete invariant for $v\in V$, where $\phi:V\to W$ is the Springer map (see $\lx@sectionsign$6.6).

Theorem 2.2.

Let $v_{0}\in V_{0}$ and $v,v^{\prime}\in V$. If $\phi(v)=\phi(v^{\prime})$ and $Y_{v_{0}}(v)\cap Y_{v_{0}}(v^{\prime})\neq\emptyset,$ then $v=v^{\prime}$.

Since $Z_{v_{0}}(v)\subset Y_{v_{0}}(v)$ for any $v_{0}\in V_{0}$ and $v\in V$, one also has

Corollary 2.3.

Let $v_{0}\in V_{0}$ and $v,v^{\prime}\in V$. If $\phi(v)=\phi(v^{\prime})$ and $Z_{v_{0}}(v)\cap Z_{v_{0}}(v^{\prime})\neq\emptyset,$ then $v=v^{\prime}$.

2.5. Subexpressions and admissible paths

Reduced decompositions for elements in $V$ and their subexpressions are introduced in [12, $\lx@sectionsign$5] and [13, $\lx@sectionsign$3, $\lx@sectionsign$4] (and see $\lx@sectionsign$6.11 for details). In particular, if $v,v^{\prime}\in V$ and if $({\bf v}^{\prime},{\bf s}^{\prime})$ is any reduced decomposition of $v^{\prime}$, then [13, Proposition 4.4] $v\leq v^{\prime}$ if and only if there exists a subexpression of $({\bf v}^{\prime},{\bf s}^{\prime})$ with final term $v$.

Let $v_{0}\in V_{0}$ and $v\in V$. We show that if $y\in Y_{v_{0}}(v)$, then every reduced decomposition of $m(y)\!\cdot\!v_{0}$ coming from a reduced word of $y$ has exactly one subexpression with final term $v$. See Proposition 7.6 for details.

For $v_{0}\in V_{0}$ and $v\in V$, we define an admissible path from $v_{0}$ to $v$ to be a pair $({\bf v},{\bf s})$, where ${\bf v}=(v_{0},v_{1},\ldots,v_{k})$ is a sequence in $V$ and ${\bf s}=(s_{1},\ldots,s_{k})$ is a sequence in $S$, such that for each $j\in[1,k]$, $v_{j}$ is a certain type of either the cross action or the monoidal action of $s_{j}$ on $v_{j-1}$ (see Definition 8.1). For such a path $({\bf v},{\bf s})$, let $y({\bf v},{\bf s})=s_{k}\cdots s_{1}\in W$. We also define the set ${\mathcal{P}}_{\rm min}(v_{0},v)$ (resp. ${\mathcal{P}}_{{\rm short}}(v_{0},v)$) of minimal (resp. shortest) paths from $v_{0}$ to $v$. We prove (see Corollary 8.8 and Corollary 8.10) that for any $v_{0}\in V_{0}$ and $v\in V$,

$$Y_{v_{0}}(v) =\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)\}$$

$$Z_{v_{0}}(v) =\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm short}(v_{0},v)\}.$$

We believe that the minimal and shortest paths defined in this paper will have other applications to the study of $K$-orbit closures in ${\mathcal{B}}$.

3. Review on $K$-orbits in ${\mathcal{B}}$, I

Following [13, $\lx@sectionsign$1.7], we review in this section the canonical Weyl group $W$ of $G$ with its set $S$ of canonical generators and the action of the monoid $M(W,S)$ on $V$. More structures on $V$ will be reviewed in $\lx@sectionsign$6.

3.1. Notation

If $Q$ is a subgroup of $G$ and $g\in G$, $Q^{g}$ denotes the subgroup $g^{-1}Qg$ of $G$. We will consider the right action of $G$ on the flag variety ${\mathcal{B}}$ by ${\mathcal{B}}\times G\to{\mathcal{B}}:(B,g)\mapsto B^{g}$ for $B\in{\mathcal{B}}$ and $g\in G$.

If a group $L$ acts on a set $X$ from the left (resp. right), we will denote by $L\backslash X$ (resp. $X/L$) the set of $L$-orbits on $X$.

Throughout the paper and unless otherwise specified, for a subset $X$ of $G$ (resp. ${\mathcal{B}}$ and ${\mathcal{B}}\times{\mathcal{B}}$), $\overline{X}$ denotes the Zariski closure of $X$ in $G$ (resp. ${\mathcal{B}}$ and ${\mathcal{B}}\times{\mathcal{B}}$).

3.2. The variety ${\mathcal{C}}$

Let ${\mathcal{C}}$ be the set of all pairs $(B,H)$, where $B\in{\mathcal{B}}$ and $H\subset B$ is a maximal torus of $G$. Then $G$ acts on ${\mathcal{C}}$ transitively from the right by

(3.1) $${\mathcal{C}}\times G\longrightarrow{\mathcal{C}}:\;\;(B,H)^{g}=(B^{g},\;H^{g}),\hskip 14.454pt(B,H)\in{\mathcal{C}},\,g\in G.$$

The stabilizer subgroup of $G$ at $(B,H)\in{\mathcal{C}}$ is $H$. Thus for each $(B,H)\in{\mathcal{C}}$, one has the $G$-equivariant identification

(3.2) $$C_{B,H}:\;\;\;H\backslash G\longrightarrow{\mathcal{C}}:\;\;\;Hg\longmapsto(B^{g},H^{g}),\hskip 14.454ptg\in G.$$

For $(B,H)\in{\mathcal{C}}$, let $N_{G}(H)$ be the normalizer of $H$ in $G$, let $W_{H}=N_{G}(H)/H$ be the Weyl group of $(G,H)$, and let $S_{B,H}$ be the set of generators of $W_{H}$ defined by the simple roots of $B$ relative to $H$.

Let $(B,H),(B^{\prime},H^{\prime})\in{\mathcal{C}}$. Let $g\in G$ be such that $B^{\prime}=B^{g}$ and $H^{\prime}=H^{g}$. Since $g$ is unique up to the left multiplication by an element in $H$, one has a well-defined isomorphism of tori

(3.3) $$T_{B,H}^{B^{\prime},H^{\prime}}:\;\;H\longrightarrow H^{\prime}:\;\;h\longmapsto g^{-1}hg,\hskip 14.454pth\in H.$$

Moreover, although the map $N_{G}(H)\to N_{G}(H^{\prime}):n\to g^{-1}ng$ depends on the choice of $g$, the group isomorphism

$$\eta_{B,H}^{B^{\prime},H^{\prime}}:\;\;W_{H}\longrightarrow W_{H^{\prime}}:\;\;nH\longmapsto g^{-1}ngH^{\prime},\hskip 14.454ptn\in N_{G}(H)$$

does not and is thus well-defined. Since $\eta_{B,H}^{B^{\prime},H^{\prime}}(S_{B,H})=S_{B^{\prime},H^{\prime}}$, one has the isomorphism $\eta_{B,H}^{B^{\prime},H^{\prime}}:(W_{H},S_{B,H})\to(W_{H^{\prime}},S_{B^{\prime},H^{\prime}})$ of Coxeter groups.

3.3. The canonical Weyl group

Let $W=({\mathcal{B}}\times{\mathcal{B}})/G$ be the set of $G$-orbits on ${\mathcal{B}}\times{\mathcal{B}}$ for the diagonal $G$-action, and let $p:{\mathcal{B}}\times{\mathcal{B}}\to W$ be the natural projection. Let $(B,H)\in{\mathcal{C}}$. Then the map

(3.4) $$\eta_{B,H}:\;\;\;W_{H}\longrightarrow W:\;\;nH\longmapsto p(B^{n},B),\hskip 14.454ptn\in N_{G}(H)$$

is bijective. It is straightforward to check that for another $(B^{\prime},H^{\prime})\in{\mathcal{C}}$,

$$\eta_{B^{\prime},H^{\prime}}^{-1}\circ\eta_{B,H}=\eta_{B,H}^{B^{\prime},H^{\prime}}:\;\;\;W_{H}\longrightarrow W_{H^{\prime}}.$$

Thus there is a well-defined group structure on $W$ such that $\eta_{B,H}:W_{H}\to W$ is a group isomorphism for every $(B,H)\in{\mathcal{C}}$. Let $S=\eta_{B,H}(S_{B,H})\subset W$ for any $(B,H)\in{\mathcal{C}}$. Then $S$ is a set of generators for $W$ and is independent of the choice of $(B,H)\in{\mathcal{C}}$. The Coxeter group $(W,S)$ is called the canonical Weyl group of $G$.

For $w\in W$, a reduced word of $w$ is a shortest expression $w=s_{1}\cdots s_{l(w)}$ of $w$ as a product of elements in $S$, and $l(w)$ is called the length of $w$.

3.4. The Bruhat order on $W$

For $w\in W$, let

$${\mathcal{O}}(w)=\{(B^{\prime},B)\in{\mathcal{B}}\times{\mathcal{B}}:\,p(B^{\prime},B)=w\}\subset{\mathcal{B}}\times{\mathcal{B}}$$

be the corresponding $G$-orbit in ${\mathcal{B}}\times{\mathcal{B}}$. The Bruhat order on $W$ is defined by

$$w\leq w^{\prime}\hskip 14.454pt\mbox{if}\hskip 14.454pt{\mathcal{O}}(w)\subset\overline{{\mathcal{O}}(w^{\prime})},\;\;\;w,w^{\prime}\in W.$$

If $w\leq w^{\prime}$ and $w\neq w^{\prime}$, we will write $w<w^{\prime}$ or $w^{\prime}>w$. It is well-known [6, Théorème 3.13] that for $w,w^{\prime}\in W$, $w^{\prime}\leq w$ if and only if there is a reduced word $w=s_{1}\cdots s_{l(w)}$ of $w$ such that $w^{\prime}=s_{i_{1}}\cdots s_{i_{p}}$ for some $1\leq i_{1}<\cdots<i_{p}\leq l(w)$.

For a subset $W_{1}$ of $W$, recall from $\lx@sectionsign$2.2 that $\min(W_{1})$ is the set of minimal elements in $W_{1}$ with respect to the Bruhat order and that $\min_{l}(W_{1})$ is the set of minimal length elements in $W_{1}$. The following Lemma 3.1 will be used in the proof of Proposition 4.5.

Lemma 3.1.

Let $W_{1}$ and $W_{2}$ be two subsets of $W$ such that $\min(W_{2})\subset W_{1}\subset W_{2}$. Then $\min(W_{1})=\min(W_{2})$ and $\min_{l}(W_{1})=\min_{l}(W_{2})$.

3.5. Orbits in ${\mathcal{B}}$ under a Borel subgroup

Let $B\in{\mathcal{B}}$. The set of $B$-orbits in ${\mathcal{B}}$ is naturally indexed by $W$. Indeed, for $w\in W$, let

(3.5) $$B(w)=\{B^{\prime}\in{\mathcal{B}}:\,p(B^{\prime},B)=w\}\subset{\mathcal{B}}.$$

Then $B(w)$ is a single $B$-orbit in ${\mathcal{B}}$, and the map $w\mapsto B(w)$ is a bijection from $W$ to the set of all $B$-orbits in ${\mathcal{B}}$.

Define $q_{B}:G\to{\mathcal{B}}$ by $q_{B}(g)=B^{g}$ for $g\in G$. For $w\in W$, let

(3.6) $$BwB=q_{B}^{-1}(B(w))=\{g\in G:p(B^{g},B)=w\}\subset G.$$

Then $BwB$ be the single $(B,B)$-double coset in $G$. Moreover, for $w,w^{\prime}\in W$,

$$w\leq w^{\prime}\hskip 14.454pt\mbox{iff}\hskip 14.454ptB(w)\subset\overline{B(w^{\prime})}\hskip 14.454pt\mbox{iff}\hskip 14.454ptBwB\subset\overline{Bw^{\prime}B}.$$

3.6. The monoidal action of $M(W,S)$ on $V$

Our references for this subsection are [12, $\lx@sectionsign$4] and [13, $\lx@sectionsign$3]. The monoid $M(W,S)$ associated to the Coxeter group $(W,S)$ is $M(W,S)=\{m(w):w\in W\}$ as a set, with the monoidal product given by

(3.7) $$m(s)m(w)=\begin{cases}m(sw),&\;\;\;\mbox{if}\;\;sw>w\\ m(w),&\;\;\;\mbox{if}\;\;sw<w\end{cases},\hskip 14.454pts\in S,\;w\in W.$$

Alternatively, let $B\in{\mathcal{B}}$, and let $BwB=q_{B}^{-1}(B(w))\subset G$ for $w\in W$ as in (3.6). Define $\ast:W\times W\to W$ such that

(3.8) $$\overline{B(w_{1}\ast w_{2})B}=\overline{(Bw_{1}B)(Bw_{2}B)},\hskip 14.454ptw_{1},w_{2}\in W.$$

Then $\ast$ is a monoidal product on $W$, independent of the choice of $B\in{\mathcal{B}}$, and $(W,\ast)\to M(W,S):w\mapsto m(w)$ is an isomorphism of monoids [6, Proposition 3.18].

Let $q_{K}:G\to G/K$ be the natural projection. Let $B\in{\mathcal{B}}$, and for $v\in V$, let

$$BvK=q_{B}^{-1}(K(v))\subset G.$$

Then $q_{K}(BvK)$, being a $B$-orbit in $G/K$, is irreducible for each $v\in V$. For $w\in W$ and $v\in V$, since $q_{K}((BwB)(BvK))\subset G/K$ is irreducible and is a finite union of $B$-orbits in $G/K$, there is a unique element in $V$, denoted by $m(w)\!\cdot\!v$, such that $\overline{q_{K}((BwB)(BvK)}=\overline{q_{K}(B(m(w)\!\cdot\!v)K)}$, where $\,\bar{}\,$ denotes the Zariski closure in $G/K$. Consequently, one has, for any $w\in W$ and $v\in V$,

(3.9) $$\overline{B(m(w)\!\cdot\!v)K}=\overline{(BwB)(BvK)}.$$

Lemma 3.2.

[12, $\lx@sectionsign$4] The map

(3.10) $$M(W,S)\times V\longrightarrow V:\;\;\;(m(w),v)\longmapsto m(w)\!\cdot\!v,\hskip 14.454ptw\in W,v\in V,$$

is independent of the choice of $B\in{\mathcal{B}}$ and defines a monoidal action of $M(W,S)$ on $V$.

Proof.

Let $B\in{\mathcal{B}},g_{0}\in G$, and $B^{\prime}=B^{g_{0}}$. It follows from definitions that for any $w\in W$ and $v\in V$, $B^{\prime}wB^{\prime}=g_{0}^{-1}BwBg_{0}$ and $B^{\prime}vK=g_{0}^{-1}BvK$. Thus, the map in (3.10) is independent of the choice of $B$. Choosing any $B\in{\mathcal{B}}$ and using the isomorphism $(W,\ast)\to M(W,S)$ of monoids, one sees from (3.8) and (3.9) that the map in (3.10) defines an action of $M(W,S)$ on $V$.

Q.E.D.

Lemma 3.3.

For any $B\in{\mathcal{B}},v\in V$, and $w\in W$, one has

$$\overline{B(m(w)\!\cdot\!v)K}=\overline{BwB}\;\;\overline{BvK}.$$

Proof.

Let $w=s_{1}s_{2}\cdots s_{l}$ be a reduced word, and for each $1\leq j\leq l$, let $P_{j}=B\cup Bs_{j}B$ so that $P_{j}$ is a parabolic subgroup of $G$. Then $\overline{BwB}=P_{1}P_{2}\cdots P_{j}$ (see, for example [6, Théorème 3.13]). By repeatedly applying [6, Lemma 2.12], one sees that $\overline{BwB}\;\;\overline{BvK}$ is closed in $G$. Thus $\overline{BwB}\;\;\overline{BvK}=\overline{(BwB)(BvK)}=\overline{B(m(w)\!\cdot\!v)K}$.

Q.E.D.

Recall that $V_{0}=\{v_{0}\in V:K(v_{0})\subset{\mathcal{B}}\;\mbox{is closed}\}$. For a subset $X$ of ${\mathcal{B}}$, let

$$X\cdot K=\{B^{k}:\;B\in X,k\in K\}.$$

Lemma 3.4.

Let $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. Then for any $w\in W$,

$$\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)\cdot K}=\overline{B(w)}\cdot K.$$

Proof.

Since $K(v_{0})$ is closed in ${\mathcal{B}}$, the double coset $Bv_{0}K=BK=q_{B}^{-1}(K(v_{0}))$ is closed in $G$. By Lemma 3.3, one has

$$\overline{B(m(w)\!\cdot\!v_{0})K}=\overline{BwB}\;\overline{BK}=\overline{BwB}\;K.$$

Applying $q_{B}:G\to{\mathcal{B}}$, one proves Lemma 3.4.

Q.E.D.

Lemma 3.5.

If $w,w^{\prime}\in W$ and $v,v^{\prime}\in V$ are such that $w\leq w^{\prime}$, $v\leq v^{\prime}$, then

$$m(w)\!\cdot\!v\leq m(w^{\prime})\!\cdot\!v\hskip 14.454pt\mbox{and}\hskip 14.454ptm(w)\!\cdot\!v\leq m(w)\!\cdot\!v^{\prime}.$$

Proof.

This is immediate from Lemma 3.3.

Q.E.D.

4. The geometrical and combinatorial definitions of $W_{v_{0}}(v)$

4.1. The two definitions of $W_{v_{0}}(v)$

Recall from $\lx@sectionsign$2.1 that for $v_{0}\in V_{0}$ and $v\in V$, the subset $W_{v_{0}}(v)$ of $W$ is defined by

$$W_{v_{0}}(v)=\{w\in W:K(v)\cap\overline{B(w)}\neq\emptyset\},$$

where $B$ is any Borel subgroup of $G$ contained in $K(v_{0})$.

Lemma 4.1.

For any $v_{0}\in V$ and $v\in V$, the set $W_{v_{0}}(v)$ is independent of the choice of $B\in K(v_{0})$.

Proof.

Let $B,B^{\prime}\in K(v_{0})$ and let $B^{\prime}=B^{k}$ for some $k\in K$. Let $w\in W$. Then $B^{\prime}(w)=\{B_{1}^{k}:B_{1}\in B(w)\}$. Thus $K(v)\cap\overline{B^{\prime}(w)}\neq\emptyset$ if and only if $K(v)\cap\overline{B(w)}\neq\emptyset$.

Q.E.D.

We now have the following combinatorial interpretation of $W_{v_{0}}(v)$.

Lemma 4.2.

For any $v_{0}\in V_{0}$ and $v\in V$, one has

$$W_{v_{0}}(v)=\{w\in W:\;v\leq m(w)\!\cdot\!v_{0}\}.$$

Proof.

Let $w\in W$. By Lemma 3.4 and by the definition of the Bruhat order on $V$, $v\leq m(w)\!\cdot\!v_{0}$ if and only if $K(v)\subset\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)}\cdot K$, which is equivalent to $K(v)\cap\overline{B(w)}\neq\emptyset$.

Q.E.D.

4.2. Properties of $W_{v_{0}}(v)$

Recall from $\lx@sectionsign$2.2 that for $v_{0}\in V_{0}$ and $v\in V$, $Y_{v_{0}}(v)$ is the set of minimal elements in $W_{v_{0}}(v)$ with respect to the Bruhat order on $W$.

Lemma 4.3.

Let $v_{0}\in V_{0}$, $v\in V$, and $w\in W$. Then $w\in W_{v_{0}}(v)$ if and only if $w\geq y$ for some $y\in Y_{v_{0}}(v)$.

Proof.

It follows from the definition of $Y_{v_{0}}(v)$ that $w\geq y$ for some $y\in Y_{v_{0}}(v)$. Conversely, assume that $w\geq y$ for some $y\in Y_{v_{0}}(v)$. Then $K(v)\cap\overline{B(y)}\neq\emptyset$ and $\overline{B(y)}\subset\overline{B(w)}$. Thus $K(v)\cap\overline{B(w)}\neq\emptyset$.

Q.E.D.

Lemma 4.4.

For $v_{0}\in V_{0}$ and $v\in V$, one has $W_{v_{0}}(v)=W$ if and only if $v=v_{0}$.

Proof.

Let $1$ be the identity element of $W$. It is clear that $1\in W_{v_{0}}(v_{0})$, so $W_{v_{0}}(v_{0})=W$ by Lemma 4.3. Assume that $v\in V$ is such that $W_{v_{0}}(v)=W$. Then $1\in W_{v_{0}}(v)$, so $v\leq v_{0}$. Since $K(v_{0})\subset{\mathcal{B}}$ is closed, we have $v=v_{0}$.

Q.E.D.

4.3. The set $W_{v_{0}}^{\prime}(v)$

Fix $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. For $v\in V$, let

(4.1) $$W_{v_{0}}^{\prime}(v)=\{w\in W:\;K(v)\cap B(w)\neq\emptyset\}.$$

By the proof of Lemma 4.1, $W_{v_{0}}^{\prime}(v)$ is independent of the choice of $B$ in $K(v_{0})$. One also sees from the definition of $B(w)$ in (3.5) that for any $v_{0}\in V_{0}$ and $v\in V$,

(4.2) $$W_{v_{0}}^{\prime}(v)=\{w\in W:\;w=p(B^{\prime},B)\;\mbox{for some}\;B^{\prime}\in K(v),\,B\in K(v_{0})\}.$$

The following Proposition 4.5 expresses $Y_{v_{0}}(v)$ and $Z_{v_{0}}(v)$ in terms of $W_{v_{0}}^{\prime}(v)$.

Proposition 4.5.

For any $v_{0}\in V_{0}$ and $v\in V$, one has

(4.3) $$Y_{v_{0}}(v)\subset W_{v_{0}}^{\prime}(v)\subset W_{v_{0}}(v).$$

Consequently, $Y_{v_{0}}(v)=\min(W_{v_{0}}^{\prime}(v))$ and $Z_{v_{0}}(v)=\min_{l}(W_{v_{0}}^{\prime}(v))$.

Proof.

Clearly $W_{v_{0}}^{\prime}(v)\subset W_{v_{0}}(v)$. Let $w\in Y_{v_{0}}(v)$. Then $K(v)\cap\overline{B(w)}\neq\emptyset$ and $K(v)\cap B(w^{\prime})=\emptyset$ for any $w^{\prime}\in W$ such that $w^{\prime}<w$. Thus $K(v)\cap B(w)\neq\emptyset$ and $w\in W_{v_{0}}^{\prime}(v)$. By Lemma 3.1, $Y_{v_{0}}(v)=\min(W_{v_{0}}^{\prime}(v))$, and $Z_{v_{0}}(v)=\min_{l}(W_{v_{0}}^{\prime}(v))$.

Q.E.D.

Example 4.6.

Let $\tilde{G}=G\times G$ and let

$$\tilde{\theta}:\;\;\tilde{G}\longrightarrow\tilde{G}:\;\;\;\tilde{\theta}(g_{1},g_{2})=(g_{2},g_{1}),\hskip 14.454pt(g_{1},g_{2})\in\tilde{G}.$$

Then the fixed point subgroup $\tilde{K}$ of $\tilde{\theta}$ in $\tilde{G}$ is $\tilde{K}=\{(g,g):g\in G\}$, so the set $\tilde{V}$ of $\tilde{K}$-orbits in $\tilde{{\mathcal{B}}}={\mathcal{B}}\times{\mathcal{B}}$ is $W$. For $w\in W$, let $\tilde{K}(w)={\mathcal{O}}(w)$, where we recall that ${\mathcal{O}}(w)$ is the $G$-orbit in $\tilde{{\mathcal{B}}}$ for the diagonal action. Then the only $\tilde{v}_{0}\in\tilde{V}=W$ such that $\tilde{K}(\tilde{v}_{0})$ is closed in $\tilde{{\mathcal{B}}}$ is $\tilde{v}_{0}=1$, the identity element of $W$. Let $\tilde{W}=W\times W$, and for $w\in W=\tilde{V}$, let $\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)\subset\tilde{W}_{\tilde{v}_{0}}(w)\subset\tilde{W}$ be defined as in (2.2) and (4.1) but for the pair $(\tilde{G},\tilde{\theta})$. Then it is easy to see that, for any $w\in W=\tilde{V}$,

$$\tilde{W}_{\tilde{v}_{0}}(w)=\{(w_{1},w_{2})\in W\times W:\;w\leq w_{1}\ast w_{2}^{-1}\},$$

where recall from $\lx@sectionsign$3.6 that $\ast$ is the monoidal product on $W$, while

$$\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)=\{(w_{1},w_{2})\in W\times W:\;BwB\subset Bw_{1}Bw_{2}^{-1}B\},$$

where $B$ is any Borel subgroup of $G$. The set $\tilde{W}_{\tilde{v}_{0}}^{\prime}(w)$ can be computed by choosing any reduced word for $w_{1}$ and using inductively the fact that, for $s\in S$ and $u\in W$, $BsBuB=BsuB$ if $su>u$ and $BsBuB=BsuB\cup BuB$ if $su<u$. See [6, Remarques 3.19]. Moreover, for any $w\in W$,

$${\rm min}(\tilde{W}_{\tilde{v}_{0}}(w)) ={\rm min}(\tilde{W}_{\tilde{v}_{0}}^{\prime}(w))$$

$$=\{(w_{1},w_{2})\in W\times W:\;w=w_{1}w_{2}^{-1},\;l(w)=l(w_{1})+l(w_{2})\},$$

and all elements in ${\rm min}(\tilde{W}_{\tilde{v}_{0}}(w))$ have the same length, namely $l(w)$.

4.4. The element $m(w)\!\cdot\!v_{0}$

Fix $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. For $w\in W$, let

(4.4) $$V_{w}=\{v\in V:\;K(v)\cap B(w)\neq\emptyset\}.$$

Then $V_{w}$ depends only on $w$ and not on the choice of $B\in K(v_{0})$.

Proposition 4.7.

Let $v_{0}\in V_{0}$, and let $B\in K(v_{0})$. Then for every $w\in W$,

1) $B(w)\cap K(m(w)\!\cdot\!v_{0})$ is dense in $B(w)$,

2) $m(w)\!\cdot\!v_{0}$ is the unique maximal element in $V_{w}$ with respect to the Bruhat order on $V$.

Proof.

Since $B(w)$ is irreducible and since ${\mathcal{B}}=\bigsqcup_{v\in V}K(v)$, there exists a unique $v_{w}\in V$ such that $B(w)\cap K(v_{w})$ is dense in $B(w)$. Moreover, if $v\in V_{w}$, then

$$\emptyset\neq B(w)\cap K(v)\subset B(w)\subset\overline{B(w)}=\overline{B(w)\cap K(v_{w})}\subset\overline{K(v_{w})},$$

and so $v\leq v_{w}$. This shows that $v_{w}$ is the unique maximal element in $V_{w}$ with respect to the Bruhat order on $V$. It remains to show that $v_{w}=m(w)\!\cdot\!v_{0}$.

Since $w\in W_{v_{0}}(v_{w})$, $v_{w}\leq m(w)\!\cdot\!v_{0}$. By Lemma 3.4, $\overline{K(m(w)\!\cdot\!v_{0})}=\overline{B(w)\cdot K}$, so $K(m(w)\!\cdot\!v_{0})\cap(B(w)\cdot K)\neq\emptyset$, and hence $K(m(w)\!\cdot\!v_{0})\cap B(w)\neq\emptyset$. Thus $m(w)\!\cdot\!v_{0}\in V_{w}$ and $m(w)\!\cdot\!v_{0}\leq v_{w}$. Hence $v_{w}=m(w)\!\cdot\!v_{0}$.

Q.E.D.

5. Intersections of $G_{0}$-orbits and $B$-orbits

In this section, we assume that ${\bf k}={\mathbb{C}}$ and that $G_{0}=\{g\in G:\tau(g)=g\}$ is a real form of $G$, where $\tau$ is an anti-holomorphic involution on $G$. Let $\sigma$ be a Cartan involution of $G$ commuting with $\tau$, and let $\theta=\sigma\tau$. Let $K=G^{\theta}$ and $K_{0}=G_{0}\cap K$. Then $K_{0}$ is a maximal compact subgroup of $G_{0}$ and $K$ is a complexification of $K_{0}$.

If $X$ is a subset of ${\mathcal{B}}$, $\overline{X}$ in this section will denote the closure of $X$ in ${\mathcal{B}}$ in the classical topology. If $X=K(v)$ or $X=B(w)$, where $v\in V$, $B$ is any Borel subgroup of $G$, and $w\in W$, the closure of $X$ in the classical topology coincides with that in the Zariski topology.

5.1. The Matsuki duality

By Matsuki duality [8, 10] [13, $\lx@sectionsign$6], for each $v\in V$, there exists a unique $G_{0}$-orbit $G_{0}(v)$ in ${\mathcal{B}}$ such that $G_{0}(v)\cap K(v)$ is a single $K_{0}$-orbit. The map $v\mapsto G_{0}(v)$ gives an identification of the set $V={\mathcal{B}}/K$ of $K$-orbits in ${\mathcal{B}}$ with the set ${\mathcal{B}}/G_{0}$ of $G_{0}$-orbits in ${\mathcal{B}}$. For $v\in V$, let $K_{0}(v)=G_{0}(v)\cap K(v).$

The following Proposition 5.1 is proved in [10, Corollary 1.4] [13, Theorem 6.4.5].

Proposition 5.1.

For $v,v^{\prime}\in V$, one has

$$v\leq v^{\prime}\hskip 14.454pt\mbox{iff}\hskip 14.454ptG_{0}(v^{\prime})\subset\overline{G_{0}(v)}\hskip 14.454pt\mbox{iff}\hskip 14.454ptG_{0}(v)\cap K(v^{\prime})\neq\emptyset.$$

Let $v_{0}\in V_{0}$ and fix $B\in K(v_{0})$. Recall the map $q_{B}:G\rightarrow{\mathcal{B}}:g\mapsto B^{g}$ for $g\in G$. For $v\in V$, let

$$BvG_{0}=q_{B}^{-1}(G_{0}(v)),\hskip 14.454ptBvK_{0}=q_{B}^{-1}(K_{0}(v)),$$

and recall that $BvK=q_{B}^{-1}(K(v))$. Let $B_{K}=K\cap B$.

Lemma 5.2.

1) $K=K_{0}B_{K}$ and $G_{0}K=G_{0}B_{K}$.

2) For any $v\in V$, one has $BvG_{0}K=\bigsqcup_{v^{\prime}\in V,v^{\prime}\geq v}Bv^{\prime}K$.

Proof.

1) Let $U=\{g\in G:\sigma(g)=g\}$ and $H=B\cap\sigma(B)$. Then $U$ is a compact real form of $G$ and $H$ a maximal torus of $G$. Let $A=\{h\in H:\sigma(h)=h^{-1}\}$ and let $N$ be the uniradical of $B$. Then one has the Iwasawa decomposition $G=UAN$ of $G$, and $U$, $A$, and $N$ are all $\theta$-invariant. Let $k\in K$ and write $k=uan$ with $u\in U,a\in A$, and $n\in N$. It follows from $\theta(k)=k$ that $u\in K_{0}$ and $an\in B_{K}$. Thus $K=K_{0}B_{K}$. It follows from $K_{0}\subset G_{0}$ that $G_{0}K=G_{0}B_{K}$.

2) Let $v\in V$. Then $BvG_{0}K$ is a union of $(B,K)$-double cosets, and for $v^{\prime}\in V$,

$$Bv^{\prime}K\subset BvG_{0}K\hskip 14.454pt\mbox{iff}\hskip 14.454ptBv^{\prime}K\cap BvG_{0}\neq\emptyset,$$

which, by Proposition 5.1, is equivalent to $v\leq v^{\prime}$.

Q.E.D.

5.2. Proof of Theorem 2.1

Let $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. Let $v\in V$ and $w\in W$. We must prove that

$$G_{0}(v)\cap B(w)\neq\emptyset\hskip 14.454pt\mbox{iff}\hskip 14.454ptv\leq m(w)\!\cdot\!v_{0}.$$

Let $BwB=q_{B}^{-1}(B(w))\subset G$. Then, by definition, $G_{0}(v)\cap B(w)\neq\emptyset$ if and only if $(BvG_{0})\cap(BwB)\neq\emptyset$. Since $B_{K}\subset B$, this last statement is equivalent to $(BvG_{0}B_{K})\cap(BwB)\neq\emptyset$, which, by 1) of Lemma 5.2, is in turn equivalent to $(BvG_{0}K)\cap(BwB)\neq\emptyset$. By 2) of Lemma 5.2, $G_{0}(v)\cap B(w)\neq\emptyset$ if and only if $v\leq v^{\prime}$ for some $v^{\prime}\in V_{w}$, where $V_{w}=\{v^{\prime}\in V:\;K(v^{\prime})\cap B(w)\neq\emptyset\}$.

By Proposition 4.7, $m(w)\!\cdot\!v_{0}$ is the unique maximal element in $V_{w}$ with respect to the Bruhat order on $V$. Thus $G_{0}(v)\cap B(w)\neq\emptyset$ if and only if $v\leq m(w)\!\cdot\!v_{0}$.

This finishes the proof of Theorem 2.1.

Corollary 5.3.

Let $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. Then for any $v\in V$ and $w\in W$, the following are equivalent:

1) $G_{0}(v)\cap B(w)\neq\emptyset$;

2) $\overline{G_{0}(v)}\cap B(w)\neq\emptyset$;

3) $G_{0}(v)\cap\overline{B(w)}\neq\emptyset$;

4) $\overline{G_{0}(v)}\cap\overline{B(w)}\neq\emptyset$;

5) $K(v)\cap\overline{B(w)}\neq\emptyset$;

6) $K_{0}(v)\cap\overline{B(w)}\neq\emptyset$.

Proof.

Clearly 1) implies 4). Assume 4). Then there exist $v_{1}\in V$ and $w_{1}\in W$ with $v_{1}\geq v$ and $w_{1}\leq w$, such that $G_{0}(v_{1})\cap B(w_{1})\neq\emptyset$. By Theorem 2.1, $v_{1}\leq m(w_{1})\!\cdot\!v_{0}$, and by Lemma 3.5, $v\leq v_{1}\leq m(w_{1})\!\cdot\!v_{0}\leq m(w)\!\cdot\!v_{0}.$ Thus 1) holds by Theorem 2.1. Hence 1) is equivalent to 4) and consequently also equivalent to 2) and 3).

By Theorem 2.1 and Lemma 4.2, both 1) and 5) are equivalent to $v\leq m(w)\!\cdot\!v_{0}$. Since $K=K_{0}B_{K}$ by 1) of Lemma 5.2, 5) and 6) are equivalent.

Q.E.D.

Lemma 5.4.

Let $v_{0}\in V_{0}$ and let $B\in K(v_{0})$. Then for any $v\in V$ and $w\in W$, the following are equivalent:

1) $K(v)\cap B(w)\neq\emptyset$;

2) $K_{0}(v)\cap B(w)\neq\emptyset$.

Proof.

It follows from $K=K_{0}B_{K}$ that 1) and 2) are equivalent.

Q.E.D.

6. Review on $K$-orbits in ${\mathcal{B}}$, II

Keeping the notation as in $\lx@sectionsign$3, we now review the canonical definitions of more structures on the set $V$ of $K$-orbits in ${\mathcal{B}}$, notably the cross action, the Springer map, root types, and reduced decompositions. All the results in this section, except Proposition 6.8 in $\lx@sectionsign$6.8, can be found in [12, 13].

6.1. The canonical maximal torus $H_{\rm can}$ and the $W$-action on $H_{\rm can}$

Recall from $\lx@sectionsign$3.2 that ${\mathcal{C}}$ is the variety of all pairs $(B,H)$, where $B\in{\mathcal{B}}$ and $H\subset B$ is a maximal torus of $G$. Let $\widetilde{{\mathcal{C}}}=\{(B,H,h)\in{\mathcal{C}}\times G:(B,H)\in{\mathcal{C}},\,h\in H\}$, and let $G$ act on $\widetilde{{\mathcal{C}}}$ by

$$(B,H,h)^{g}=(B^{g},H^{g},g^{-1}hg),\hskip 14.454pt(B,H,h)\in\widetilde{{\mathcal{C}}},\;g\in G.$$

Let $H_{\rm can}=\widetilde{{\mathcal{C}}}/G$ be the set of $G$-orbits in $\widetilde{{\mathcal{C}}}$, and let $\widetilde{{\mathcal{C}}}\to H_{\rm can}:(B,H,h)\to[B,H,h]$ be the canonical projection. Then for every $(B,H)\in{\mathcal{C}}$, one has the bijective map

$$T_{B,H}:\;H\longrightarrow H_{\rm can}:\;\;h\longmapsto[B,H,h],\hskip 14.454pth\in H.$$

Since $T_{B^{\prime},H^{\prime}}^{-1}\circ T_{B,H}=T_{B,H}^{B^{\prime},H^{\prime}}:H\to H^{\prime}$ for any $(B,H),(B^{\prime},H^{\prime})\in{\mathcal{C}}$, where $T_{B,H}^{B^{\prime},H^{\prime}}$ is the isomorphism of tori in (3.3), $H_{\rm can}$ has a well-defined structure of a torus, such that $T_{B,H}:H\to H_{\rm can}$ is an isomorphism of tori for every $(B,H)\in{\mathcal{C}}$. We will call $H_{\rm can}$ the canonical maximal torus of $G$.

The canonical Weyl group $W$ acts on $H_{\rm can}$ through the action of $W_{H}$ on $H$ via the identifications $\eta_{B,H}^{-1}:W\to W_{H}$ and $T_{B,H}^{-1}:H_{\rm can}\to H$ for any $(B,H)\in{\mathcal{C}}$, and the action is independent of the choice of $(B,H)\in{\mathcal{C}}$. If $(B,H),(B^{\prime},H)\in{\mathcal{C}}$, then

(6.1) $$p(B^{\prime},B)([B,H,h])=[B^{\prime},H,h],\hskip 14.454pth\in H.$$

One can also take (6.1) as the definition of the canonical action of $W$ on $H_{\rm can}$.

6.2. The canonical root system of $G$

Let $X_{\rm can}$ be the character group of $H_{\rm can}$. For $(B,H)\in{\mathcal{C}}$, let $X_{H}$ be the character group of $H$, and let $R_{B,H}:X_{H}\to X_{\rm can}$ be the isomorphism induced by $T_{B,H}^{-1}:H_{\rm can}\to H$. The canonical sets of roots, positive roots, and simple roots of $G$, are the subsets of $X_{\rm can}$, respectively defined by

$$\Delta=R_{B,H}(\Delta_{H}),\hskip 14.454pt\Delta^{+}=R_{B,H}(\Delta^{+}_{B,H}),\hskip 14.454pt\Gamma=R_{B,H}(\Gamma_{B,H}),$$

where $(B,H)\in{\mathcal{C}}$, and $\Delta_{H}\supset\Delta^{+}_{B,H}\supset\Gamma_{B,H}$ are the subsets of the character group $X_{H}$ of $H$ consisting, respectively, of the roots of $(G,H)$, the positive roots, and the simple roots determined by $B$. It is easy to check that the sets $\Gamma\subset\Delta^{+}\subset\Delta\subset X_{\rm can}$ are independent of the choice of $(B,H)\in{\mathcal{C}}$.

For $\alpha\in\Delta$, choose any $(B,H)\in{\mathcal{C}}$, let $\alpha_{H}=R_{B,H}^{-1}(\alpha)\in\Delta_{H}$, let $s_{\alpha_{H}}\in W_{H}$ be the reflection defined by $\alpha_{H}$, and let $s_{\alpha}=\eta_{B,H}(s_{\alpha_{H}})\in W$. Then $s_{\alpha}$ is independent of the choice of $(B,H)\in{\mathcal{C}}$. Moreover, $S=\{s_{\alpha}:\alpha\in\Gamma\}$.

6.3. The automorphism $\theta$ on $H_{\rm can}$, $\Gamma$, and $W$

In this subsection, let $\theta$ be any automorphism of $G$, not necessarily of order $2$. Then one has the well-defined map

$$\theta:\;H_{\rm can}\longrightarrow H_{\rm can}:\;\;[B,H,h]\longmapsto[\theta(B),\theta(H),\theta(h)],\hskip 14.454pt(B,H,h)\in\widetilde{{\mathcal{C}}}.$$

If $(B,H)\in{\mathcal{C}}$, then, by definition, $\theta=T_{\theta(B),\theta(H)}\circ\theta|_{H}\circ T_{B,H}^{-1}$, where $\theta|_{H}:H\to\theta(H)$. Thus $\theta$ is an automorphism of $H_{\rm can}$.

It is clear from the definition that if $\theta$ is inner, i.e., if there exists $g_{1}\in G$ such that $\theta(g)=g_{1}gg_{1}^{-1}$ for $g\in G$, then $\theta$ induces the identity automorphism of $H_{\rm can}$.

We will use the same letter to denote the induced action of $\theta$ on the set $\Delta$ of canonical roots. It follows from the definitions that $\theta(\Delta^{+})=\Delta^{+}$ and $\theta(\Gamma)=\Gamma$.

The following map, again denoted by $\theta$, is easily seen to be well-defined:

$$\theta:\;W\longrightarrow W:\;\;\;p(B_{1},B_{2})\longmapsto p(\theta(B_{1}),\theta(B_{2})),\hskip 14.454ptB_{1},B_{2}\in{\mathcal{B}},$$

Choose any $(B,H)\in{\mathcal{C}}$, and let $\theta_{W_{H}}:W_{H}\to W_{\theta(H)}$ be the group isomorphism induced by $\theta|_{N_{G}(H)}:N_{G}(H)\to N_{G}(\theta(H))$. It follows from definitions that

$$\theta=\eta_{\theta(B),\theta(H)}\circ\theta_{W_{H}}\circ\eta_{B,H}^{-1}:\;\;W\longrightarrow W.$$

Thus $\theta$ is an automorphism of $W$. It also follows that $\theta(s_{\alpha})=s_{\theta(\alpha)}$ for all $\alpha\in\Delta$.

6.4. The identification $\gamma:{\mathcal{C}}_{\theta}/K\to V$

Let ${\mathcal{T}}^{\theta}$ be the set of all $\theta$-stable maximal tori in $G$. The following Lemma 6.1 is proved in [13, Proposition 1.2.1] and [16, Corollary 4.4]. See also [12, 1.4(b)].

Lemma 6.1.

Every $B\in{\mathcal{B}}$ contains some $H\in{\mathcal{T}}^{\theta}$, and if $H_{1},H_{2}\in{\mathcal{T}}^{\theta}$ are both contained in $B$, then $H_{1}=k^{-1}H_{2}k$ for some $k\in B\cap K$.

Let ${\mathcal{C}}_{\theta}=\{(B,H)\in{\mathcal{C}}:H\in{\mathcal{T}}^{\theta}\}$ and let $K$ act on ${\mathcal{C}}_{\theta}$ by (3.1). It follows from Lemma 6.1 that the well-defined map

$$\gamma:\;{\mathcal{C}}_{\theta}/K\longrightarrow V={\mathcal{B}}/K:\;\mbox{{\it K}-orbit of}\;(B,H)\;\mbox{in}\;{\mathcal{C}}\longmapsto\mbox{{\it K}-orbit of}\;B\;\mbox{in}\;{\mathcal{B}},$$

is a bijection ([13, Proposition 1.2.1] and [16, Corollary 4.4]).

6.5. The cross action of $W$ on $V$

We follow [12, $\lx@sectionsign$2] and [13, $\lx@sectionsign$1.7]. See also [4]. For any $(B,H)\in{\mathcal{C}}$ and $w\in W$, there is a unique $(B^{\prime},H)\in{\mathcal{C}}$ such that $w=p(B^{\prime},B)$. Define

(6.2) $$W\times{\mathcal{C}}\longrightarrow{\mathcal{C}}:\;\;w\cdot(B,H)=(B^{\prime},H),\hskip 14.454pt(B,H),(B^{\prime},H)\in{\mathcal{C}},\,w=p(B^{\prime},B).$$

If we fix $(B,H)\in{\mathcal{C}}$, then under the identifications $\eta_{B,H}:W_{H}\to W$ and $C_{B,H}:H\backslash G\to{\mathcal{C}}:Hg\mapsto(B^{g},H^{g})$ for $g\in G$, the map in (6.2) becomes

$$W_{H}\times H\backslash G\longrightarrow H\backslash G:\;\;\;(nH,Hg)\longmapsto Hng,\hskip 14.454ptn\in N_{G}(H),\,g\in G,$$

which is a left action of $W_{H}$ on $H\backslash G$. Thus (6.2) is indeed a left action of $W$ on ${\mathcal{C}}$. Moreover, the $W$-action commutes with the right action of $G$ on ${\mathcal{C}}$ given in (3.1).

By (6.2), ${\mathcal{C}}_{\theta}\subset{\mathcal{C}}$ is $W$-invariant. Thus one has a well-defined action of $W$ on ${\mathcal{C}}_{\theta}/K$. Identifying $V$ with ${\mathcal{C}}_{\theta}/K$ via $\gamma$, one gets a left action of $W$ on $V$, which is called the cross action of $W$ on $V$ and will be denoted by

$$W\times V\longrightarrow V:\;\;\;(w,v)\longmapsto w\!\cdot\!v,\hskip 14.454ptw\in W,\,v\in V.$$

6.6. The Springer map $\phi:V\to W$

The Springer map $\phi:V\to W$, introduced by Springer in [16], is defined (see [12, Remark 1.8] and [13, Proposition 1.7.1]) by

(6.3) $$\phi(v)=p(B,\theta(B)),\hskip 14.454ptv\in V,\;B\in K(v).$$

The element $\phi(v)\in W$ for $v\in V$ is an important invariant of the $K$-orbit $K(v)\subset{\mathcal{B}}$. Recall that $V_{0}$ consists of all $v\in V$ such that $K(v)$ is closed in ${\mathcal{B}}$. Let $1$ be the identity element of $W$. The following Proposition 6.2 is from [13, Proposition 1.4.2] and [12, Proposition 2.5].

Proposition 6.2.

1) For $v\in V$, $v\in V_{0}$ if and only if $\phi(v)=1$.

2) If $v,v^{\prime}\in V$ are such that $\phi(v)=\phi(v^{\prime})$, then $v$ and $v^{\prime}$ are in the same $W$-orbit.

6.7. Using standard pairs

Let $(B,H)\in{\mathcal{C}}$ be a standard pair, i.e., $\theta(B)=B$ and $\theta(H)=H$. Then for any $g\in G$, $(B^{g},H^{g})\in{\mathcal{C}}_{\theta}$ if and only if $g\theta(g)^{-1}\in N_{G}(H)$, and for such a $g\in G$,

$$p(B^{g},\,\theta(B^{g}))=p(B^{g},\,B^{\theta(g)})=p(B^{g\theta(g)^{-1}},B)=\eta_{B,H}(g\theta(g)^{-1}H)\in W.$$

Letting ${\mathcal{V}}_{H}=\{g\in G:\,g\theta(g)^{-1}\in N_{G}(H)\}$, one thus has the identification

(6.4) $$H\backslash{\mathcal{V}}_{H}/K\longrightarrow V:\;\;\;HgK\longmapsto\mbox{the}\;\mbox{{\it K}-orbit in}\;{\mathcal{B}}\;\mbox{through}\;B^{g}.$$

Under the identification of $V$ with $H\backslash{\mathcal{V}}_{H}/K$ in (6.4) and that of $W$ with $W_{H}$ by $\eta_{B,H}$, the Springer map $\phi:V\to W$ becomes $\phi(HgK)=g\theta(g)^{-1}H$ for $g\in{\mathcal{V}}_{H}$, and the action of $W$ on $V$ becomes $(nH)\!\cdot\!(HgK)=HngK$ for $n\in N_{G}(H)$ and $g\in{\mathcal{V}}_{H}$.

In [12, 13, 16], most of the structures on $V$ are introduced and their properties proved by using standard pairs. For example, the following Lemma 6.3 immediately follows from the the identification (6.4) (see also [12, Lemma 2.1]).

Lemma 6.3.

One has $\phi(w\!\cdot\!v)=w\phi(v)\theta(w)^{-1}$ for any $w\in W$ and $v\in V$.

6.8. The closed $K$-orbits in ${\mathcal{B}}$

Let $W^{\theta}=\{w\in W:\theta(w)=w\}$. By Lemma 6.3,

(6.5) $$\{w\in W:\;w\!\cdot\!v_{0}=v_{0}^{\prime}\}=\{w\in W^{\theta}:\;w\!\cdot\!v_{0}=v_{0}^{\prime}\},\hskip 14.454pt\forall\,v_{0},v_{0}^{\prime}\in V_{0}.$$

By Proposition 6.2, $W^{\theta}$ acts transitively on $V_{0}$.

For the rest of this subsection, we assume that $K$ is connected. We will relate the sets in (6.5) for $v_{0},v_{0}^{\prime}\in V_{0}$ with the canonical Weyl group $W_{K}$ of $K$. Proposition 6.8 will be used in $\lx@sectionsign$9.1, where we determine $Y_{v_{0}}(v_{0}^{\prime})$ for every $v_{0},v_{0}^{\prime}\in V_{0}$.

Lemma 6.4.

Let $v_{0},v_{0}^{\prime}\in V_{0}$. Then for any $B\in K(v_{0})$ and $B^{\prime}\in K(v_{0}^{\prime})$, there exists $H\in{\mathcal{T}}^{\theta}$ such that $H\subset B\cap B^{\prime}$.

Proof.

Since $K(v_{0})$ and $K(v_{0}^{\prime})$ are closed, it follows from [13, Theorem 1.4.3] (see also [16, Corollary 6.6]) that $B$ and $B^{\prime}$ are $\theta$-stable. Thus, $B\cap B^{\prime}$ is $\theta$-stable and contains a maximal torus of $G$. Hence, by [18, Theorem 7.5], $B\cap B^{\prime}$ contains a $\theta$-stable maximal torus of $G$.

Q.E.D.

Let $v_{0},v_{0}^{\prime}\in V_{0}$, and consider the restriction $p:K(v_{0}^{\prime})\times K(v_{0})\rightarrow W$. By definition,

$$p(K(v_{0}^{\prime})\times K(v_{0}))=\{w\in W:w=p(B^{\prime},B)\,\mbox{for some}\;B^{\prime}\in K(v_{0}^{\prime}),\,B\in K(v_{0})\}.$$

Lemma 6.5.

For any $v_{0},v_{0}^{\prime}\in V_{0}$, one has

$$p(K(v_{0}^{\prime})\times K(v_{0}))=\{w\in W:w\!\cdot\!v_{0}=v_{0}^{\prime}\}.$$

Proof.

By the definition of the $W$-action, $\{w\in W:w\!\cdot\!v_{0}=v_{0}^{\prime}\}\subset p(K(v_{0}^{\prime})\times K(v_{0}))$. Suppose that $w=p(B^{\prime},B)$ for some $B\in K(v_{0})$ and $B^{\prime}\in K(v_{0}^{\prime})$. By Lemma 6.4 and by the definition of the $W$-action on $V$, $w\!\cdot\!v_{0}=v_{0}^{\prime}$.

Q.E.D.

Let ${\mathcal{B}}_{K}$ be the variety of all Borel subgroups of $K$ and let $K$ act on ${\mathcal{B}}_{K}$ by $(B_{K})^{k}:=k^{-1}B_{K}k$ for $B_{K}\in{\mathcal{B}}_{K}$ and $k\in K$.

Lemma 6.6.

For any $v_{0}\in V_{0}$, the map

$${\mathcal{I}}_{v_{0}}:\;\;\;K(v_{0})\longrightarrow{\mathcal{B}}_{K}:\;\;\;B\longmapsto B\cap K,\hskip 14.454ptB\in K(v_{0})$$

is a $K$-equivariant isomorphism.

Proof.

By [11, 5.1] (see also [13, Page 113]), $B\cap K\in{\mathcal{B}}_{K}$ for every $B\in K(v_{0})$. Thus ${\mathcal{I}}_{v_{0}}$ is well-defined. It is clear that ${\mathcal{I}}_{v_{0}}$ is $K$-equivariant. To show that ${\mathcal{I}}_{v_{0}}$ is an isomorphism, we show that ${\mathcal{I}}_{v_{0}}$ is bijective and that its inverse is an isomorphism from ${\mathcal{B}}_{K}$ to $K(v_{0})$. Fix $B\in K(v_{0})$ and identify $(B\cap K)\backslash K\cong{\mathcal{B}}_{K}$ via

$$(B\cap K)\backslash K\longrightarrow{\mathcal{B}}_{K}:\;\;\;(B\cap K)k\longmapsto(B\cap K)^{k},\hskip 14.454ptk\in K.$$

Consider the action map $\eta:K\to K(v_{0}):k\mapsto B^{k}$, $k\in K$. Then the morphism $\tilde{\eta}:(B\cap K)\backslash K\to K(v_{0})$ induced by $\eta$ is the inverse of ${\mathcal{I}}_{v_{0}}$. By [5, Proposition 6.7 and Corollary 6.1], $\tilde{\eta}$ is an isomorphism if $\eta$ is separable.

Let ${\mathfrak{b}},{\mathfrak{k}},$ and ${\mathfrak{g}}$ be the Lie algebras of $B,K$, and $G$ respectively, and let $d\theta:{\mathfrak{g}}\to{\mathfrak{g}}$ be the differential of $\theta$. Since ${\rm char}({\bf k})\neq 2$, $\theta$ is semisimple on $G$ and $B$ [15, Section 5.4]. By [15, Theorem 5.4.4(ii)], ${\mathfrak{k}}={\mathfrak{g}}^{d\theta}=\{x\in{\mathfrak{g}}:d\theta(x)=x\}$, and the Lie algebra of $B\cap K$ coincides with ${\mathfrak{b}}\cap{\mathfrak{k}}$. Applying [5, Proposition 6.12] to the quotient morphism $G\to B\backslash G:g\to B^{g},g\in G$, one sees that $\eta$ is separable.

Q.E.D.

Identify $W_{K}=({\mathcal{B}}_{K}\times{\mathcal{B}}_{K})/K$ for the diagonal action of $K$ on ${\mathcal{B}}_{K}\times{\mathcal{B}}_{K}$. For $x\in W_{K}$, let ${\mathcal{O}}_{K}(x)\subset{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}$ be the corresponding $K$-orbit in ${\mathcal{B}}_{K}\times{\mathcal{B}}_{K}$. Let $\leq_{K}$ denote the Bruhat order on $W_{K}$.

Let $v_{0},v_{0}^{\prime}\in V_{0}$, and let

$$\mathcal{J}_{v_{0},v_{0}^{\prime}}=({\mathcal{I}}_{v_{0}^{\prime}}\times{\mathcal{I}}_{v_{0}})^{-1}:\;\;{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}\longrightarrow K(v_{0}^{\prime})\times K(v_{0}).$$

For $x\in W_{K}$, let ${\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)$ be the single $K$-orbit in $K(v_{0}^{\prime})\times K(v_{0})$ given by

(6.6) $${\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)=\mathcal{J}_{v_{0},v_{0}^{\prime}}({\mathcal{O}}_{K}(x))\subset K(v_{0}^{\prime})\times K(v_{0}).$$

One then has the well-defined map

(6.7) $$I_{v_{0},v_{0}^{\prime}}:\;\;\;W_{K}\longrightarrow W:\;\;\;x\longmapsto p\left({\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x)\right),\hskip 14.454ptx\in W_{K}.$$

Let $1$ be the identity element in $W_{K}$. By definition,

$${\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(1)=\{(B^{\prime},B)\in K(v_{0}^{\prime})\times K(v_{0}):\;B^{\prime}\cap K=B\cap K\}.$$

Definition 6.7.

For $v_{0},v_{0}^{\prime}\in V_{0}$, let $y_{v_{0},v_{0}^{\prime}}\in W^{\theta}$ be given by

(6.8) $$y_{v_{0},v_{0}^{\prime}}=I_{v_{0},v_{0}^{\prime}}(1)=p(B^{\prime},B),\hskip 14.454pt\forall(B^{\prime},B)\in{\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(1).$$

Proposition 6.8.

Let $v_{0},v_{0}^{\prime}\in V_{0}$.

1) The map $I_{v_{0},v_{0}^{\prime}}$ is a bijection from $W_{K}$ onto $\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}$;

2) If $x_{1},x_{2}\in W_{K}$ are such that $x_{1}\leq_{K}x_{2}$, then $I_{v_{0},v_{0}^{\prime}}(x_{1})\leq I_{v_{0},v_{0}^{\prime}}(x_{2})$;

3) $y_{v_{0},v_{0}^{\prime}}\in W^{\theta}$ is the unique minimal element in the set $\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}$ with respect to the Bruhat order on $W$.

Proof.

1) By Lemma 6.5 and (6.5), the image of $I_{v_{0},v_{0}^{\prime}}$ is $\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}$. To show that $I_{v_{0},v_{0}^{\prime}}$ is injective, assume that $B,B_{1}\in K(v_{0})$ and $B^{\prime},B_{1}^{\prime}\in K(v_{0}^{\prime})$ are such that $p(B^{\prime},B)=p(B_{1}^{\prime},B_{1})\in W$. We must show that $(B^{\prime},B)$ and $(B_{1}^{\prime},B_{1})$ are in the same $K$-orbit in $K(v_{0}^{\prime})\times K(v_{0})$ for the diagonal action of $K$. Without loss of generality, we may assume that $B_{1}=B$, and we need to show that $B^{\prime}=(B_{1}^{\prime})^{k}$ for some $k\in B\cap K$. Let $H,H_{1}\in{\mathcal{T}}^{\theta}$ be such that $H\subset B\cap B^{\prime}$ and $H_{1}\subset B\cap B_{1}^{\prime}$. By Lemma 6.1, there exists $k\in B\cap K$ such that $H=(H_{1})^{k}$. Thus $H\subset B\cap B^{\prime}\cap(B_{1}^{\prime})^{k}$. By the assumption, $p(B^{\prime},B)=p(B_{1}^{\prime},B)=p((B_{1}^{\prime})^{k},B^{k})=p((B_{1}^{\prime})^{k},B)$. Thus $B^{\prime}=(B_{1}^{\prime})^{k}$.

2) Suppose that $x_{1},x_{2}\in W_{K}$ are such that $x_{1}\leq_{K}x_{2}$. For $i=1,2$, let $w_{i}=I_{v_{0},v_{0}^{\prime}}(x_{i})\in W$, so that ${\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x_{i})\subset{\mathcal{O}}(w_{i})$, where recall from $\lx@sectionsign$3.4 that ${\mathcal{O}}(w_{i})$ is the $G$-orbit in ${\mathcal{B}}\times{\mathcal{B}}$ corresponding to $w_{i}\in W$. For a subset $X$ of ${\mathcal{B}}_{K}\times{\mathcal{B}}_{K}$ (resp. of ${\mathcal{B}}\times{\mathcal{B}}$), let $\overline{X}^{{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}}$ (resp. $\overline{X}^{{\mathcal{B}}\times{\mathcal{B}}}$) denote the Zariski closure of $X$ in ${\mathcal{B}}_{K}\times{\mathcal{B}}_{K}$ (resp. in ${\mathcal{B}}\times{\mathcal{B}}$). Since $\mathcal{J}_{v_{0},v_{0}^{\prime}}:{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}\to{\mathcal{B}}\times{\mathcal{B}}$ is a morphism, one has

$${\mathcal{O}}_{K,v_{0},v_{0}^{\prime}}(x_{1}) =\mathcal{J}_{v_{0},v_{0}^{\prime}}({\mathcal{O}}_{K}(x_{1}))\subset\mathcal{J}_{v_{0},v_{0}^{\prime}}\left(\overline{{\mathcal{O}}_{K}(x_{1})}^{{\mathcal{B}}_{K}\times{\mathcal{B}}_{K}}\right)\subset\overline{\mathcal{J}_{v_{0},v_{0}^{\prime}}({\mathcal{O}}_{K}(x_{1}))}^{{\mathcal{B}}\times{\mathcal{B}}}$$

$$\subset\overline{{\mathcal{O}}(w_{2})}^{{\mathcal{B}}\times{\mathcal{B}}}.$$

Thus ${\mathcal{O}}(w_{1})\cap\overline{{\mathcal{O}}(w_{2})}^{{\mathcal{B}}\times{\mathcal{B}}}\neq\emptyset$, and hence $w_{1}\leq w_{2}$.

3) follows directly from 1) and 2).

Q.E.D.

Remark 6.9.

For $v_{0}\in V_{0}$, let $W_{K}(v_{0})=\{w\in W:w\!\cdot\!v_{0}=v_{0}\}$. Then for $v_{0},v_{0}^{\prime}\in V_{0}$, the set $\{w\in W^{\theta}:\,w\!\cdot\!v_{0}=v_{0}^{\prime}\}$ coincides with the coset $y_{v_{0},v_{0}^{\prime}}W_{K}(v_{0})$ in $W^{\theta}$. It is easy to see that $I_{v_{0},v_{0}}:W_{K}\to W_{K}(v_{0})$ is a group isomorphism (see, for example, [12, Proposition 2.8]). Hence, by 3) of Proposition 6.8, every coset in $W^{\theta}/W_{K}(v_{0})$ has a unique minimal length representative. In case $K$ is disconnected, this last assertion is no longer true in the case when $G=PGL(4)$ and $K$ has connected component of the identity equal to the image of $GL(2)\times GL(2)$ in $G$.

6.9. The involution $\theta_{v}:H_{\rm can}\to H_{\rm can}$

Let $v\in V$. Choose any $(B,H)\in{\mathcal{C}}_{\theta}$ such that $B\in K(v)$, and let $\theta|_{H}:H\to H$ be the restriction of $\theta$ to $H$. Define

$$\theta_{v}:=T_{B,H}\circ\theta|_{H}\circ T_{B,H}^{-1}:\;H_{\rm can}\longrightarrow H_{\rm can}:\;\;[B,H,h]\longmapsto[B,H,\theta(h)],\hskip 14.454pth\in H.$$

For another $(B^{\prime},H^{\prime})\in{\mathcal{C}}_{\theta}$ such that $B^{\prime}\in K(v)$, there exists $k\in K$ such that $B^{\prime}=B^{k}$ and $H^{\prime}=H^{k}$, and for any $h\in H$, $[B,H,h]=[B^{k},H^{k},k^{-1}hk]\in H_{\rm can}$, and

$$[B^{k},H^{k},\theta(k^{-1}hk)]=[B^{k},H^{k},k^{-1}\theta(h)k]=[B,H,\theta(h)].$$

Thus $\theta_{v}:H_{\rm can}\to H_{\rm can}$ is independent of the choice of $(B,H)\in{\mathcal{C}}_{\theta}$. By (6.1),

(6.9) $$\theta_{v}=\phi(v)\theta:\;\;\;H_{\rm can}\longrightarrow H_{\rm can}.$$

The induced involution on the set $\Delta$ of canonical roots of $G$ (see $\lx@sectionsign$6.2) will also be denoted by $\theta_{v}$.

6.10. The subsets $p(s_{\alpha},v)$ and root types

Let $\alpha\in\Gamma$. A parabolic subgroup of $G$ is said to be of type $\alpha$ if it is of the form $B\cup Bs_{\alpha}B$ for some $B\in{\mathcal{B}}$. Let ${\mathcal{P}}_{\alpha}$ be the variety of all parabolic subgroups of $G$ of type $\alpha$. The action of $G$ on ${\mathcal{P}}_{\alpha}$ by conjugation is transitive, and one has the $G$-equivariant surjective morphism

$$\pi_{\alpha}:\;\;{\mathcal{B}}\longrightarrow{\mathcal{P}}_{\alpha}:\;\;\pi_{\alpha}(B)=B\cup(Bs_{\alpha}B),\hskip 14.454ptB\in{\mathcal{B}}.$$

For $v\in V$, let

$$p(s_{\alpha},v)=\{v^{\prime}\in V:\;K(v^{\prime})\subset\pi_{\alpha}^{-1}(\pi_{\alpha}(K(v))\}.$$

It is well-known (see, for example, [13, $\lx@sectionsign$2.4] and [14]) that for each $\alpha\in\Gamma$ and $v\in V$, the subset $p(s_{\alpha},v)$ of $V$ has either one or two or three elements, depending on the type of $\alpha$ relative to $v$, and that $m(s_{\alpha})\!\cdot\!v$ is the unique maximal element in $p(s_{\alpha},v)$ with respect to the Bruhat order on $V$.

The case analysis of $p(s_{\alpha},v)$ and the definition of the type of $\alpha$ relative to $v$ given in [13, $\lx@sectionsign$2.4] and [14] make use of a standard pair $(B_{0},H_{0})\in{\mathcal{C}}$, but the results are independent of the choice of $(B_{0},H_{0})$. Based on the results from [13, $\lx@sectionsign$2.4] and [14], we give the equivalent definitions of root types in Definition 6.10 and summarize the results on $p(s_{\alpha},v)$ from [13, $\lx@sectionsign$2.4] and [14] in the following Proposition 6.11.

Definition 6.10.

Let $v\in V$. An $\alpha\in\Gamma$ is said to be imaginary (resp. real, complex) for $v$ if $\theta_{v}(\alpha)=\alpha$ (resp. $\theta_{v}(\alpha)=-\alpha,\;\theta_{v}(\alpha)\neq\pm\alpha$). A simple imaginary root $\alpha$ for $v\in V$ is said to be compact if $m(s_{\alpha})\!\cdot\!v=v$ and non-compact if $m(s_{\alpha})\!\cdot\!v\neq v$. A simple non-compact imaginary root $\alpha$ is said to be cancellative if $s_{\alpha}\!\cdot\!v=v$ and non-cancellative if $s_{\alpha}\!\cdot\!v\neq v$ (see [17, $\lx@sectionsign$2.4]). A simple real root $\alpha$ for $v\in V$ is said to be cancellative if $p(s_{\alpha},v)$ has two elements and non-cancellative if $p(s_{\alpha},v)$ has three elements. We will use the following notation.

$$I^{c}_{v} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is compact imaginary for}\,v\},$$

$$I_{v}^{n,=} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is non-compact imaginary and cancellative for}\,v\},$$

$$I_{v}^{n,\neq} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is non-compact imaginary and non-cancellative for}\,v\},$$

$$R_{v}^{=} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is real and cancellative for}\,v\},$$

$$R_{v}^{\neq} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is real and non-cancellative for}\,v\},$$

$$C_{v}^{+} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is complex for}\,v\;\mbox{and}\;\theta_{v}(\alpha)\in\Delta^{+}\},$$

$$C_{v}^{-} =\{\alpha\in\Gamma:\;\alpha\;\mbox{is complex for}\,v\;\mbox{and}\;\theta_{v}(\alpha)\in-\Delta^{+}\}.$$

We also set $I_{v}^{n}=I_{v}^{n,=}\cup I_{v}^{n,\neq}$, $I_{v}=I_{v}^{c}\cup I_{v}^{n}$, and $R_{v}=R_{v}^{=}\cup R_{v}^{\neq}$.

Proposition 6.11.

Let $v\in V$ and $\alpha\in\Gamma$.

Case 1), $\alpha\in I_{v}^{c}$. Then $p(s_{\alpha},v)=\{v\}$.

Case 2), $\alpha\in I_{v}^{n,=}$. Then $s_{\alpha}\!\cdot\!v=v\neq m(s_{\alpha})\!\cdot\!v$, and $p(s_{\alpha},v)=\{v,\;m(s_{\alpha})\!\cdot\!v\}$. Moreover, $\alpha\in R_{m(s_{\alpha})\cdot v}^{=}$.

Case 3), $\alpha\in I_{v}^{n,\neq}$. Then $v$, $s_{\alpha}\!\cdot\!v$, and $m(s_{\alpha})\!\cdot\!v$ are pair-wise distinct, and $p(s_{\alpha},v)=\{v,\;s_{\alpha}\!\cdot\!v,\;m(s_{\alpha})\!\cdot\!v\}$. Moreover, $\alpha\in I_{s_{\alpha}\cdot v}^{n,\neq}$ and $\alpha\in R_{m(s_{\alpha})\cdot v}^{\neq}$.

Case 4), $\alpha\in R_{v}^{=}.$ Then there exists $v^{\prime}\in p(s_{\alpha},v)$, $v^{\prime}\neq v$, such that $s_{\alpha}\!\cdot\!v^{\prime}=v^{\prime},\,v=m(s_{\alpha})\!\cdot\!v^{\prime}=s_{\alpha}\!\cdot\!v$, and $p(s_{\alpha},v)=\{v^{\prime},\;v\}$. Moreover, $\alpha\in I_{v^{\prime}}^{n,=}$.

Case 5), $\alpha\in R_{v}^{\neq}$. Then there exists $v^{\prime}\in p(s_{\alpha},v)$ such that $v^{\prime},\,s_{\alpha}\!\cdot\!v^{\prime}$ and $v=m(s_{\alpha})\!\cdot\!v^{\prime}=s_{\alpha}\!\cdot\!v$ are pair-wise distinct, and $p(s_{\alpha},v)=\{v^{\prime},\,s_{\alpha}\!\cdot\!v^{\prime},\,v\}$. Moreover, $\alpha\in I_{v^{\prime}}^{n,\neq}\cap I_{s_{\alpha}\cdot v^{\prime}}^{n,\neq}$.

Case 6), $\alpha\in C_{v}^{+}.$ Then $m(s_{\alpha})\!\cdot\!v=s_{\alpha}\!\cdot\!v\neq v$, and $p(s_{\alpha},v)=\{v,\,m(s_{\alpha})\!\cdot\!v\}$. Moreover, $\alpha\in C^{-}_{m(s_{\alpha})\cdot v}$.

Case 7), $\alpha\in C_{v}^{-}.$ Then $m(s_{\alpha})\!\cdot\!v=v\neq s_{\alpha}\!\cdot\!v$, and $p(s_{\alpha},v)=\{v,\,s_{\alpha}\!\cdot\!v\}$. Moreover, $\alpha\in C^{+}_{s_{\alpha}\cdot v}$.

Lemma 6.12.

[13, Page 122] Let $v,v^{\prime}\in V$ and $\alpha\in\Gamma$. Then $v^{\prime}\in p(s_{\alpha},v)$ if and only if $m(s_{\alpha})\!\cdot\!v^{\prime}=m(s_{\alpha})\!\cdot\!v$. Moreover, $p(s_{\alpha},v^{\prime})=p(s_{\alpha},v)$ for all $v^{\prime}\in p(s_{\alpha},v).$

Lemma 6.13.

[13, $\lx@sectionsign$3.2] For $\alpha\in\Gamma$ and $v\in V$, $m(s_{\alpha})\!\cdot\!v\neq v$ if and only if $\alpha\in C_{v}^{+}\cup I_{v}^{n}$. If $\alpha\in C_{v}^{+}$, then $\phi(m(s_{\alpha})\!\cdot\!v)=s_{\alpha}\phi(v)s_{\theta(\alpha)}$, and $l(\phi(m(s_{\alpha})\!\cdot\!v))=l(\phi(v))+2$. If $\alpha\in I_{v}^{n}$, then $\phi(m(s_{\alpha})\!\cdot\!v)=s_{\alpha}\phi(v)=\phi(v)s_{\theta(\alpha)}>\phi(v)$ and $\phi(s_{\alpha}\!\cdot\!v)=\phi(v)$.

6.11. Reduced decompositions and subexpressions

We refer to [12, $\lx@sectionsign$5-7] and [13, $\lx@sectionsign$4] for more detail on this subsection.

Definition 6.14.

[13, Definition 3.2.3] Let $v\in V$. A reduced decomposition of $v$ is a pair $({\bf v},{\bf s})$, where ${\bf v}=(v_{0},v_{1},\ldots,v_{k})$ is a sequence in $V$ and ${\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})$ is a sequence of simple reflections in $W$, such that $v_{0}\in V_{0}$, $v_{k}=v$, and for each $j\in[1,k]$,

$$\alpha_{j}\in C_{v_{j-1}}^{+}\cup I_{v_{j-1}}^{n}\;\;\;\mbox{and}\;\;\;v_{j}=m(s_{\alpha_{j}})\!\cdot\!v_{j-1}.$$

The integer $k$ is called the length of the reduced decomposition $({\bf v},{\bf s})$.

Every $v\in V$ has a reduced decomposition and all reduced decompositions of $v$ have the same length, which will be denoted by $l(v)$ and called the length of $v$ (see [13, 3.2 and Page 113]).

Definition 6.15.

[13, Definition 4.3] Let $v\in V$ and let $({\bf v}=(v_{0},v_{1},\ldots,v_{k}),\,{\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}}))$ be a reduced decomposition for $v$. A subexpression of $({\bf v},{\bf s})$ is a sequence ${\bf u}=(u_{0},u_{1},\ldots,u_{k})$ in $V$ such that $u_{0}=v_{0}$ and one of the following holds for each $j\in[1,k]$:

Case 1), $u_{j}=u_{j-1}$;

Case 2), $\alpha_{j}\in C_{u_{j-1}}^{+}\cup I_{u_{j-1}}^{n}$ and $u_{j}=m(s_{\alpha_{j}})\!\cdot\!u_{j-1}$;

Case 3), $\alpha_{j}\in I_{u_{j-1}}^{n,\neq}$ and $u_{j}=s_{\alpha_{j}}\!\cdot\!u_{j-1}$.

In this case, $u_{k}$ is called the final term of the subexpression ${\bf u}=(u_{0},u_{1},\ldots,u_{k})$.

Proposition 6.16.

[13, Proposition 4.4] Let $v,v^{\prime}\in V$ and let $({\bf v},{\bf s})$ be a reduced decomposition for $v$. Then $v^{\prime}\leq v$ if and only if there exists a subexpression of $({\bf v},{\bf s})$ with final term $v^{\prime}$.

7. Analysis on $Y_{v_{0}}(v)$ and proof of Theorem 2.2

7.1. The set $Y_{v_{0}}$

For $v_{0}\in V_{0}$, let $Y_{v_{0}}=\bigcup_{v\in V}Y_{v_{0}}(v)$. Recall that $l$ denotes both the length function on $W$ and the length function on $V$.

Lemma 7.1.

Let $v_{0}\in V_{0}$ and $y\in W$. Then $y\in Y_{v_{0}}$ if and only if $l(m(y)\!\cdot\!v_{0})=l(y)$.

Proof.

It is clear that $l(m(y)\!\cdot\!v_{0})\leq l(y)$ for any $y\in W$. Assume first that $y\in Y_{v_{0}}(v)$ for some $v\in V$. If $l(m(y)\!\cdot\!v_{0})<l(y)$, then there exists $y_{1}<y$ such that $m(y_{1})\!\cdot\!v_{0}=m(y)\!\cdot\!v_{0}$, so $y_{1}\in W_{v_{0}}(v)$, which is a contradiction. Thus $l(m(y)\!\cdot\!v_{0})=l(y)$. Conversely, assume that $l(m(y)\!\cdot\!v_{0})=l(y)$. Let $v=m(y)\!\cdot\!v_{0}$. Then $y\in Y_{v_{0}}(v)$.

Q.E.D.

Lemma-Notation 7.2.

Let $y\in W$. If $y=s_{\alpha_{k}}\cdots s_{\alpha_{1}}$ is a reduced word of $y$, the sequence ${\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})$ is also called a reduced word of $y^{-1}$. Let

$$R(y^{-1})=\{{\bf s}:\;{\bf s}\;\mbox{is a reduced word of}\;y^{-1}\}.$$

Let $v_{0}\in V_{0}$ and $y\in Y_{v_{0}}$. For ${\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})\in R(y^{-1})$, let

(7.1) $${\bf v}_{v_{0}}({\bf s})=(v_{0},\;m(y_{1})\!\cdot\!v_{0},\;\cdots,\;m(y_{k})\!\cdot\!v_{0}),$$

where $y_{j}=s_{\alpha_{j}}\cdots s_{\alpha_{1}}$ for $j\in[1,k]$. Then $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ is a reduced decomposition for $m(y)\!\cdot\!v_{0}$, which will be called the reduced decomposition of $m(y)\!\cdot\!v_{0}$ associated to ${\bf s}$.

Proof.

If $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ is not a reduced decomposition for $m(y)\!\cdot\!v_{0}$, then $l(m(y)\!\cdot\!v_{0})<l(y)$, and so $y\not\in Y_{v_{0}}$ by Lemma 7.1.

Q.E.D.

7.2. Local analysis of $Y_{v_{0}}(v)$, Part I

Recall the monoidal operation $\ast$ on $W$ in $\lx@sectionsign$3.6.

Lemma 7.3.

Let $v_{0}\in V_{0}$, $v\in V$, and $\alpha\in\Gamma$.

1) If $w\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$, then $w\in W_{v_{0}}(v^{\prime})$ for all $v^{\prime}\in p(s_{\alpha},v)$;

2) If $w\in W_{v_{0}}(v)$, then $s_{\alpha}\ast w\in W_{v_{0}}(v^{\prime})$ for all $v^{\prime}\in p(s_{\alpha},v)$.

Proof.

1) follows from the fact that $v^{\prime}\leq m(s_{\alpha})\!\cdot\!v$ for all $v^{\prime}\in p(s_{\alpha},v)$.

2). Assume that $w\in W_{v_{0}}(v)$. Then by Lemma 3.5,

$$m(s_{\alpha})\!\cdot\!v\leq m(s_{\alpha})\!\cdot\!m(w)\!\cdot\!v_{0}=m(s_{\alpha}\ast w)\!\cdot\!v_{0}.$$

Thus $s_{\alpha}\ast w\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$. 2) now follows from 1).

Q.E.D.

The following Lemma 7.4 is the key in proving Theorem 2.2.

Lemma 7.4.

Let $v_{0}\in V_{0}$, $v\in V$, and $y\in Y_{v_{0}}(v)$. Let $\alpha\in\Gamma$ and $y^{\prime}\in W$ be such that $y=s_{\alpha}y^{\prime}>y^{\prime}$. Then there exists $u\in p(s_{\alpha},\,v)\backslash\{v,\,m(s_{\alpha})\!\cdot\!v\}$ such that $y^{\prime}\in Y_{v_{0}}(u)$. Moreover, there are three possibilities:

1) $\alpha\in C_{u}^{+}$ and $v=m(s_{\alpha})\!\cdot\!u$. In this case, $\alpha\in C_{v}^{-}$;

2) $\alpha\in I_{v}^{n}$ and $v=m(s_{\alpha})\!\cdot\!u$. In this case, $\alpha\in R_{v}$;

3) $\alpha\in I_{u}^{n,\neq}$ and $v=s_{\alpha}\!\cdot\!u$. In this case, $\alpha\in I_{v}^{n,\neq}$.

Proof.

Let $y^{\prime}=s_{\alpha_{k-1}}\cdots s_{\alpha_{1}}$ be a reduced word of $y^{\prime}$. Then

$${\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k-1}},s_{\alpha_{k}})\in R(y^{-1}),$$

where $\alpha_{k}=\alpha$. Consider the reduced decomposition $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ of $m(y)\!\cdot\!v_{0}$. By Proposition 6.16, there is a subexpression ${\bf u}=(u_{0}=v_{0},\,u_{1},,\ldots,,u_{k-1},\,u_{k})$ of $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ with final term $u_{k}=v$, and we let $u=u_{k-1}$. Since $y^{\prime}<y$ and $u\leq m(y^{\prime})\!\cdot\!v_{0}$ by Proposition 6.16, $u\neq v$ and $u\neq m(s_{\alpha})\!\cdot\!u=m(s_{\alpha})\!\cdot\!v$. By the definition of a subexpression, one has either 1), 2), or 3). By Proposition 6.11, $\alpha\in C_{v}^{-}$ in 1), $\alpha\in R_{v}$ in 2), and $\alpha\in I_{v}^{n,\neq}$ in 3).

It remains to show that $y^{\prime}\in Y_{v_{0}}(u)$. Since $u\leq m(y^{\prime})\!\cdot\!v_{0}$, $y^{\prime}\in W_{v_{0}}(u)$. Suppose that $y^{\prime}\notin Y_{v_{0}}(u)$. Let $y^{\prime\prime}\in W_{v_{0}}(u)$ be such that $y^{\prime\prime}<y^{\prime}$. By Lemma 7.3, $s_{\alpha}\ast y^{\prime\prime}\in W_{v_{0}}(v)$. Since $s_{\alpha}\ast y^{\prime\prime}\leq s_{\alpha}\ast y^{\prime}=y$ and $l(s_{\alpha}\ast y^{\prime\prime})\leq 1+l(y^{\prime\prime})<1+l(y^{\prime})=l(y)$, we have $s_{\alpha}\ast y^{\prime\prime}<y$, which is a contradiction. Thus $y^{\prime}\in Y_{v_{0}}(u)$.

Q.E.D.

7.3. Proof of Theorem 2.2

Assume that $v,v^{\prime}\in V$ are such that $\phi(v)=\phi(v^{\prime})$ and that there exists $y\in Y_{v_{0}}(v)\cap Y_{v_{0}}(v^{\prime})$. We use induction on $l(y)$ to show that $v=v^{\prime}$.

If $l(y)=0$, then $y=1$. By Lemma 4.4, $v=v^{\prime}=v_{0}$. Assume now that $l(y)\geq 1$, and choose $\alpha\in\Gamma$ such that $y^{\prime}:=s_{\alpha}y<y$. By Lemma 7.4, there exist $u\in p(s_{\alpha},v)$ and $u^{\prime}\in p(s_{\alpha},v^{\prime})$ such that $y^{\prime}\in Y_{v_{0}}(u)\cap Y_{v_{0}}(u^{\prime})$, and

$$\alpha\in\left(C_{v}^{-}\cup R_{v}\cup I_{v}^{n,\neq}\right)\cap\left(C_{v^{\prime}}^{-}\cup R_{v^{\prime}}\cup I_{v^{\prime}}^{n,\neq}\right).$$

Since $\phi(v)=\phi(v^{\prime})$, one has by (6.9) that $\theta_{v}(\alpha)=\theta_{v^{\prime}}(\alpha)$. Thus $\alpha\in C_{v}^{-}$ (resp. $R_{v},\,I_{v}^{n,\neq}$) if and only if $\alpha\in C_{v^{\prime}}^{-}$ (resp. $R_{v^{\prime}},\;I_{v^{\prime}}^{n,\neq}$). We now look at the cases separately.

Case 1): $\alpha\in C_{v}^{-}$. In this case, $\alpha\in C_{v^{\prime}}^{-}$, $v=m(s_{\alpha})\!\cdot\!u=s_{\alpha}\!\cdot\!u$, and $v^{\prime}=m(s_{\alpha})\!\cdot\!u^{\prime}=s_{\alpha}\!\cdot\!u^{\prime}$. By Lemma 6.13, $\phi(u)=s_{\alpha}\phi(v)\theta(s_{\alpha})=s_{\alpha}\phi(v^{\prime})\theta(s_{\alpha})=\phi(u^{\prime}).$ By the induction assumption, $u=u^{\prime}$. Thus $v=s_{\alpha}\!\cdot\!u=s_{\alpha}\!\cdot\!u^{\prime}=v^{\prime}$.

Case 2): $\alpha\in R_{v}$. In this case, $\alpha\in R_{v^{\prime}}$, $v=m(s_{\alpha})\!\cdot\!u$, and $v^{\prime}=m(s_{\alpha})\!\cdot\!u^{\prime}$. By Lemma 6.13, $\phi(u)=s_{\alpha}\phi(v)=s_{\alpha}\phi(v^{\prime})=\phi(u^{\prime})$. By the induction assumption, $u=u^{\prime}$. Thus $v=m(s_{\alpha})\!\cdot\!u=m(s_{\alpha})\!\cdot\!u^{\prime}=v^{\prime}$.

Case 3): $\alpha\in I_{v}^{n,\neq}$. In this case, $\alpha\in I_{v^{\prime}}^{n,\neq}$, $u=s_{\alpha}\!\cdot\!v$, and $u^{\prime}=s_{\alpha}\!\cdot\!v^{\prime}$. By Lemma 6.13, $\phi(u)=\phi(v)=\phi(v^{\prime})=\phi(u^{\prime})$. By the induction assumption, $u=u^{\prime}$. Thus $v=s_{\alpha}\!\cdot\!u=s_{\alpha}\!\cdot\!u^{\prime}=v^{\prime}$.

This finishes the proof of Theorem 2.2.

7.4. Local analysis on $Y_{v_{0}}(v)$, Part II

The following Lemma 7.5 strengthens Lemma 7.4 and completes the local analysis on the sets $Y_{v_{0}}(v)$.

Lemma 7.5.

The element $u$ in Lemma 7.4 is unique. If 2) of Lemma 7.4 occurs and if $\alpha\in I_{u}^{n,\neq}$, then $y\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)$. If 3) of Lemma 7.4 occurs, then $y\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$.

Proof.

The only case where $u$ might not be unique is when 2) in Lemma 7.4 occurs and when $\alpha\in I_{u}^{n,\neq}$. Assume this is the case. Since $\phi(u)=\phi(s_{\alpha}\!\cdot\!u)$, by Theorem 2.2, $y^{\prime}$ can not be in both $Y_{v_{0}}(u)$ and $Y_{v_{0}}(s_{\alpha}\!\cdot\!u)$, so the choice of $u$ is unique. Moreover, by 1) of Lemma 7.3, $y\in W_{v_{0}}(s_{\alpha}\!\cdot\!u)$. Let $y_{1}\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)$ be such that $y_{1}\leq y$. By 2) of Lemma 7.3, $s_{\alpha}\ast y_{1}\in W_{v_{0}}(v)$. Since $s_{\alpha}\ast y_{1}\leq s_{\alpha}\ast y=y$, one has $s_{\alpha}\ast y_{1}=y$. If $s_{\alpha}y_{1}>y_{1}$, then $s_{\alpha}y_{1}=s_{\alpha}\ast y_{1}=y$, so $y_{1}=s_{\alpha}y=y^{\prime}$, contradicting the fact that $y^{\prime}$ can not be in both $Y_{v_{0}}(u)$ and $Y_{v_{0}}(s_{\alpha}\!\cdot\!u)$. Thus $s_{\alpha}y_{1}<y_{1}$, and hence $y=s_{\alpha}\ast y_{1}=y_{1}\in Y_{v_{0}}(s_{\alpha}\!\cdot\!u)$.

Assume now that 3) of Lemma 7.4 occurs. Then $y=s_{\alpha}y^{\prime}\in W_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$ by 2) of Lemma 7.3. Let $y_{2}\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$ be such that $y_{2}\leq y$. By 1) of Lemma 7.3, $y_{2}\in W_{v_{0}}(v)$. Thus $y=y_{2}\in Y_{v_{0}}(m(s_{\alpha})\!\cdot\!v)$.

Q.E.D.

7.5. Elements of $Y_{v_{0}}(v)$ and subexpressions

We now prove the following key property of $y\in Y_{v_{0}}(v)$ in terms of subexpressions of reduced decompositions of $m(y)\!\cdot\!v_{0}$.

Proposition 7.6.

Let $v_{0}\in V_{0},v\in V$ and $y\in Y_{v_{0}}(v)$. Then for any reduced word ${\bf s}=(s_{\alpha_{1}},\ldots,s_{\alpha_{k}})$ of $y^{-1}$, there is exactly one subexpression ${\bf u}=(v_{0},u_{1},\ldots,u_{k})$ of the reduced decomposition $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ of $m(y)\!\cdot\!v_{0}$ that has final term $v$. Moreover, for any $1\leq j\leq k$, $u_{j}\neq u_{j-1}$ and $s_{\alpha_{j}}s_{\alpha_{j-1}}\cdots s_{\alpha_{1}}\in Y_{v_{0}}(u_{j})$.

Proof.

By Proposition 6.16, there is a subexpression ${\bf u}=(u_{0}=v_{0},u_{1},\ldots,u_{k-1},u_{k})$ of the reduced decomposition $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ of $m(y)\!\cdot\!v_{0}$ with final term $v$. Letting $\alpha=\alpha_{k}$ and $y^{\prime}=s_{\alpha_{k-1}}\cdots s_{\alpha_{1}}$, one knows from Lemma 7.4 and Lemma 7.5 that $y^{\prime}\in Y_{v_{0}}(u_{k-1})$ and that $u_{k-1}$ is uniquely determined by the quadruple $(v_{0},v,y,\alpha)$. Proceeding inductively, one proves Proposition 7.6.

Q.E.D.

Notation 7.7.

For $v_{0}\in V_{0}$, $v\in V$, $y\in Y_{v_{0}}(v)$, and ${\bf s}\in R(y^{-1})$, the unique subexpression of the reduced decomposition $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ of $m(y)\!\cdot\!v_{0}$ (see Lemma-Notation 7.2) with final term $v$ will be denoted by ${\bf u}_{v_{\scriptscriptstyle 0},v}({\bf s})$.

8. Admissible paths and the sets $Y_{v_{0}}(v)$ and $Z_{v_{0}}(v)$

8.1. Admissible paths

Proposition 7.6 leads naturally to the following notion of admissible paths.

Definition 8.1.

Fix $v_{0}\in V_{0}$.

i) For $v\in V$, an admissible path from $v_{0}$ to $v$ is a pair $({\bf v},{\bf s})$, where ${\bf v}=(v_{0},v_{1},\ldots,v_{k}=v)$ is a sequence in $V$, and ${\bf s}=(s_{\alpha_{1}},s_{\alpha_{2}},\ldots,s_{\alpha_{k}})$ is a sequence of simple reflections in $W$, and $k\geq 1$, such that one of the following holds for each $j\in[1,k]$:

1) $\alpha_{j}\in C_{v_{j-1}}^{+}\cup I_{v_{j-1}}^{n}$ and $v_{j}=m(s_{\alpha_{j}})\!\cdot\!v_{j-1}$;

2) $\alpha_{j}\in I_{v_{j-1}}^{n,\neq}$ and $v_{j}=s_{\alpha_{j}}\!\cdot\!v_{j-1}$.

The pair $({\bf v}=\{v_{0}\},\,{\bf s}=\emptyset)$ is called the trivial admissible path from $v_{0}$ to $v_{0}$ with the understanding that $k=0$ in this case. The set of all admissible paths from $v_{0}$ to $v$ is denoted by ${\mathcal{P}}(v_{0},v)$.

ii) For $({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)$ as in i), the number $k$ is called the length of $({\bf v},{\bf s})$ and denoted by $l({\bf v},{\bf s})$. We also set $y_{0}({\bf v},{\bf s})=1\in W$, $y_{j}({\bf v},{\bf s})=s_{\alpha_{j}}s_{\alpha_{j-1}}\cdots s_{\alpha_{1}}$ for $j\in[1,k]$, and $y({\bf v},{\bf s})=y_{k}({\bf v},{\bf s})$.

iii) For $v\in V$, $({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)$ is said to be a shortest admissible path from $v_{0}$ to $v$ if $l({\bf v},{\bf s})\leq l({\bf v}^{\prime},{\bf s}^{\prime})$ for every $({\bf v}^{\prime},{\bf s}^{\prime})\in{{\mathcal{P}}(v_{0},v)}$. The length of a shortest admissible path from $v_{0}$ to $v$ will be denoted by $l_{v_{0}}(v)$.

Remark 8.2.

We are using the symbol $({\bf v},{\bf s})$ to denote both a reduced decomposition of an element in $V$ and an admissible path in $V$. This should cause no confusion as we will always use the modifiers “reduced decomposition” or “admissible path” in front of $({\bf v},{\bf s})$. Moreover, it is clear from the definitions that a reduced decomposition of $v\in V$ starting from $v_{0}\in V_{0}$ is an admissible path from $v_{0}$ to $v$.

Lemma 8.3.

Let $v_{0}\in V_{0}$ and $v\in V$. Let $({\bf v},{\bf s})\in{{\mathcal{P}}(v_{0},v)}$, where ${\bf s}=(s_{1},\dots s_{k})$. Then $v\leq m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}$.

Proof.

Let ${\bf v}=(v_{0},v_{1}\ldots,v_{k-1},v_{k}=v)$. Then

$$v\leq m(s_{k})\!\cdot\!v_{{k-1}}\leq\cdots\leq m(s_{k})m(s_{{k-1}})\cdots m(s_{1})\!\cdot\!v_{0}=m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}.$$

Q.E.D.

8.2. Minimal admissible paths and elements in $Y_{v_{0}}(v)$

By Definition 6.15, for $v_{0}\in V_{0},v\in V,y\in Y_{v_{0}}(v)$, and ${\bf s}\in R(y^{-1})$, the pair $({\bf u}_{v_{0},v}({\bf s}),{\bf s})$ (see Notation 7.7) is an admissible path from $v_{0}$ to $v$. In this section, we give a characterization of such admissible paths.

Definition 8.4.

Let $v_{0}\in V_{0}$ and $v\in V$. For a sequence ${\bf s}$ of simple reflections in $W$, let

$${\mathcal{P}}(v_{0},v,{\bf s})=\{({\bf v}^{\prime},{\bf s}^{\prime})\in{\mathcal{P}}(v_{0},v):\;{\bf s}^{\prime}\;\mbox{is a subsequence of}\;{\bf s}\}.$$

A path $({\bf v},{\bf s})\in{\mathcal{P}}(v_{0},v)$ is said to be minimal if $({\bf v},{\bf s})$ is the only member of ${\mathcal{P}}(v_{0},v,{\bf s})$. The set of all minimal admissible paths from $v_{0}$ to $v$ will be denoted by ${\mathcal{P}}_{\rm min}(v_{0},v)$.

Lemma 8.5.

Let $v_{0}\in V_{0}$ and $v\in V$.

1) If $y\in Y_{v_{0}}(v)$ and ${\bf s}\in R(y^{-1})$, then $({\bf u}_{v_{0},v}({\bf s}),{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)$;

2) If $({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)$ and $y=y({\bf v},{\bf s})$, then $y\in Y_{v_{0}}(v)$, ${\bf s}\in R(y^{-1})$, and $({\bf v},{\bf s})=({\bf u}_{v_{0},v}({\bf s}),{\bf s})$.

Proof.

1) Let $y\in Y_{v_{0}}(v)$ and ${\bf s}\in R(y^{-1})$. Write ${\bf s}=(s_{1},\ldots,s_{k})$, and let ${\bf s}^{\prime}=(s_{i_{1}},\ldots,s_{i_{p}})$ be a subsequence of ${\bf s}$. Suppose that $({\bf v}^{\prime},{\bf s}^{\prime})\in{{\mathcal{P}}(v_{0},v)}$. By Lemma 8.3, $v\leq m(s_{i_{p}}\ast\cdots\ast s_{i_{1}})\!\cdot\!v_{0}$. Since $s_{i_{p}}\ast\cdots\ast s_{i_{1}}\leq y$, one must have $s_{i_{p}}\ast\cdots\ast s_{i_{1}}=y$, which is possible only when ${\bf s}^{\prime}={\bf s}$. It follows that ${\bf v}^{\prime}$ is a subexpression of the reduced decomposition $({\bf v}_{v_{0}}({\bf s}),{\bf s})$ of $m(y)\!\cdot\!v_{0}$. By Proposition 7.6, ${\bf v}^{\prime}={\bf u}_{v_{0},v}({\bf s})$. This proves that $({\bf u}_{v_{0},v}({\bf s}),{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)$.

2) Let $({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)$ and let $y=y({\bf v},{\bf s})$. Let ${\bf s}=(s_{1},\ldots,s_{k})$. By Lemma 8.3, $v\leq m(s_{k}\ast\cdots\ast s_{1})\!\cdot\!v_{0}$. Suppose that $s_{k}\ast\cdots\ast s_{1}\neq s_{k}\cdots s_{1}$. Then there exists a proper subsequence ${\bf s}^{\prime}=(s_{i_{1}},\ldots,s_{i_{p}})$ of ${\bf s}$ such that ${\bf s}^{\prime}$ is a reduced word of $y^{\prime}=s_{k}\ast\cdots\ast s_{1}=s_{i_{p}}\cdots s_{i_{1}}$. By passing to a subsequence of ${\bf s}^{\prime}$ if necessary, we can assume that $y^{\prime}\in Y_{v_{0}}(v)$. Then $({\bf u}_{v_{0},v}({\bf s}^{\prime}),{\bf s}^{\prime})$ is an admissible path from $v_{0}$ to $v$ that is different from $({\bf v},{\bf s})$, contradicting the assumption on $({\bf v},{\bf s})$. Thus $y=s_{k}\ast\cdots\ast s_{1}=s_{k}\cdots s_{1}$, so ${\bf s}$ is a reduced word of $y^{-1}$. The same arguments show that $y\in Y_{v_{0}}(v)$. By Proposition 7.6, $({\bf v},{\bf s})=({\bf u}_{v_{0},v}({\bf s}),{\bf s})$.

Q.E.D.

For $v_{0}\in V_{0}$ and $v\in V$, let

$${\mathcal{Y}}_{v_{0}}(v)=\{(y,\,{\bf s}):\,y\in Y_{v_{0}}(v),\,{\bf s}\in R(y^{-1})\}.$$

The following Proposition 8.6 follows immediately from Lemma 8.5.

Proposition 8.6.

For any $v_{0}\in V_{0}$ and $v\in V$, the map

(8.1) $$\Omega:\;\;\;{\mathcal{P}}_{\rm min}(v_{0},v)\longrightarrow{\mathcal{Y}}_{v_{0}}(v):\;\;\;({\bf v},{\bf s})\longmapsto(y({\bf v},{\bf s}),\,{\bf s})$$

is bijective, with inverse given by

(8.2) $$\Omega^{-1}:\;\;\;{\mathcal{Y}}_{v_{0}}(v)\longrightarrow{\mathcal{P}}_{\rm min}(v_{0},v):\;\;\;(y,{\bf s})\longmapsto({\bf u}_{v_{0},v}({\bf s}),{\bf s}).$$

Definition 8.7.

For $v_{0}\in V_{0},v\in V$, and $(y,{\bf s})\in{\mathcal{Y}}_{v_{0}}(v)$, we will call $({\bf u}_{v_{0},v}({\bf s}),{\bf s})$ the admissible path from $v_{0}$ to $v$ associated to ${\bf s}$.

Corollary 8.8.

For any $v_{0}\in V$ and $v\in V$, one has

$$Y_{v_{0}}(v)=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)\}.$$

8.3. Shortest admissible paths and elements in $Z_{v_{0}}(v)$

Let $v_{0}\in V_{0}$ and $v\in V$. Let ${\mathcal{P}}_{\rm short}(v_{0},v)$ denote the set of all shortest admissible paths from $v_{0}$ to $v$ (see Definition 8.1). It is clear that ${\mathcal{P}}_{\rm short}(v_{0},v)\subset{\mathcal{P}}_{\rm min}(v_{0},v)$. Let

$${\mathcal{Z}}_{v_{0}}(v)=\{(z,\,{\bf s}):\,z\in Z_{v_{0}}(v),\,{\bf s}\in R(z^{-1})\}.$$

Then ${\mathcal{Z}}_{v_{0}}(v)\subset{\mathcal{Y}}_{v_{0}}(v)$.

Proposition 8.9.

For any $v_{0}\in V_{0}$ and $v\in V$, the map $\Omega$ in (8.1) restricts to a bijection between ${\mathcal{P}}_{\rm short}(v_{0},v)$ and ${\mathcal{Z}}_{v_{0}}(v)$.

Proof.

Since $l({\bf v},{\bf s})=l(y({\bf v},{\bf s}))$ for all $({\bf v},{\bf s})\in{\mathcal{P}}_{\rm min}(v_{0},v)$, Proposition 8.9 follows directly from Proposition 8.6.

Q.E.D.

Corollary 8.10.

For any $v_{0}\in V_{0}$ and $v\in V$, one has

$$Z_{v_{0}}(v)=\{y({\bf v},{\bf s}):({\bf v},{\bf s})\in{\mathcal{P}}_{\rm short}(v_{0},v)\}.$$

Example 8.11.

Let $v_{0}\in V_{0}$, and let $M(W,S)\!\cdot\!v_{0}=\{m(w)\!\cdot\!v_{0}:w\in W\}$. Suppose that $v\in M(W,S)\!\cdot\!v_{0}$. Then

(8.3) $$Z_{v_{0}}(v)=\{z\in W:\;m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}.$$

Indeed, if $w\in W_{v_{0}}(v)$, then $l(v)\leq l(m(w)\!\cdot\!v_{0})\leq l(w).$ Since $v\in M(W,S)\!\cdot\!v_{0}$, there exists $z_{0}\in W$ such that $v=m(z_{0})\!\cdot\!v_{0}$ and $l(z_{0})=l(v)$. Thus $l(v)=\min\{l(w):w\in W_{v_{0}}(v)\}$. Let $Z_{v_{0}}^{\prime}(v)$ be the set of the right hand side on (8.3). Then $Z_{v_{0}}^{\prime}(v)\subset Z_{v_{0}}(v)$. Conversely, if $z\in Z_{v_{0}}(v)$, then $l(z)=l(v)$, and it follows from $l(v)\leq l(m(z)\!\cdot\!v_{0})\leq l(z)$ that $v=m(z)\!\cdot\!v_{0}$. Thus $Z_{v_{0}}(v)\subset Z_{v_{0}}^{\prime}(v)$. Hence $Z_{v_{0}}(v)=Z_{v_{0}}^{\prime}(v)$.

We will see in $\lx@sectionsign$9.4 that it can happen that $v\in M(W,S)\!\cdot\!v_{0}$ and $Z_{v_{0}}(v)$ is a proper subset of $Y_{v_{0}}(v)$.

9. Examples

9.1. The case when $v\in V_{0}$

In this subsection, we assume that $K$ is connected. Fix $v_{0},v_{0}^{\prime}\in V_{0}$. We will consider the set $Y_{v_{0}}(v_{0}^{\prime})$.

Recall from $\lx@sectionsign$4.3 that $Y_{v_{0}}(v_{0}^{\prime})=\min(W_{v_{0}}^{\prime}(v_{0}^{\prime}))$, where

(9.1) $$W_{v_{0}}^{\prime}(v_{0}^{\prime})=\{w\in W:\;w=p(B^{\prime},B)\;\mbox{for some}\;B^{\prime}\in K(v_{0}^{\prime}),\,B\in K(v_{0})\},$$

and $p:{\mathcal{B}}\times{\mathcal{B}}\to W=({\mathcal{B}}\times{\mathcal{B}})/G$ is the natural projection. By Lemma 6.5, $W_{v_{0}}^{\prime}(v_{0}^{\prime})=\{w\in W^{\theta}:w\!\cdot\!v_{0}=v_{0}^{\prime}\}.$ Recall from Definition 6.7 the element $y_{v_{0},v_{0}^{\prime}}\in W^{\theta}$. By 3) of Proposition 6.8, $y_{v_{0},v_{0}^{\prime}}$ is the unique minimal element in $W_{v_{0}}^{\prime}(v_{0}^{\prime})$. Thus $Y_{v_{0}}(v_{0}^{\prime})=\{y_{v_{0},v_{0}^{\prime}}\}$ has one element.

9.2. The case when there is a unique closed orbit

Assume that $V_{0}=\{v_{0}\}$ has only one element. Then $M(W,S)\!\cdot\!v_{0}=V$. In this case, for every $v\in V$, one has

$$Y_{v_{0}}(v)=Z_{v_{0}}(v)=\{z\in W:\;m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}.$$

Indeed, by [12, Corollary 9.15], the Springer map $\phi:V\to W$ is injective. For any $v\in V$, since $\phi(s_{\alpha}\!\cdot\!v)=\phi(v)$ for any $\alpha\in I_{v}^{n}$, one has $I_{v}^{n,\neq}=\emptyset$, and thus for any admissible path $({\bf v},{\bf s})$ from $v_{0}$ to $v$, 2) in Definition 8.1 does not occur. Therefore if $y\in Y_{v_{0}}(v)$, then $v=m(y)\!\cdot\!v_{0}$ and $l(y)=l(v)$, and so $y\in Z_{v_{0}}(v)$.

Remark 9.1.

For any $v\in V$, and not assuming that $V_{0}$ has only one element, T. A. Springer has studied in [17] the set $\cup_{v_{0}\in V_{0}}\{z\in W:m(z)\!\cdot\!v_{0}=v,\,l(z)=l(v)\}$ as an invariant for $v$.

9.3. The Hermitian symmetric case

Assume that $G$ is simple and simply connected. By [13, Definition 5.1.1], $(G,\theta)$ is said to be of Hermitian symmetric type if the center of $K$ has positive dimension.

Assume that $(G,\theta)$ is of Hermitian symmetric type. By [13, Theorem 5.12], there exists a standard pair $(B,H)\in{\mathcal{C}}$ and a parabolic subgroup $P$ of $G$ containing $B$ such that $K$ is the unique Levi subgroup of $P$ containing $H$. Moreover, there exists $\alpha_{0}\in\Gamma$ such that $P$ is of type $J=\Gamma\backslash\{\alpha_{0}\}$. Let $v_{0}\in V$ such that $B\in K(v_{0})$. Then every $\alpha\in J$ is compact imaginary for $v_{0}$. We now study the set $Y_{v_{0}}(v)$ for every $v\in V$.

Let $W_{J}$ be the subgroup of $W$ generated by simple reflections corresponding to roots in $J$, and let $W^{J}$ be the set of minimal length representatives for $W/W_{J}$ in $W$. Then $W^{J}$ parametrizes the set of $(B,P)$-double cosets in $G$ via

$$W^{J}\ni d\longmapsto BdP:=B\eta_{B,H}^{-1}(d)P\subset G,$$

where $\eta_{B,H}:W_{H}\to W$ is given in (3.4). Since every $(B,K)$-double coset in $G$ is contained in a unique $(B,P)$-double coset, we have the well-defined surjective map

$$\nu:\;V\longrightarrow W^{J}:\;\;\;\nu(v)=d\in W^{J}\hskip 14.454pt\mbox{{\rm if}}\hskip 14.454ptBvK\subset BdP.$$

It is proved in [13, Theorem 5.2.5] that the map

$$\eta:\;V\longrightarrow W\times W^{J}:\;\;\eta(v)=(\phi(v),\;\nu(v))$$

is injective.

Proposition 9.2.

Let $(G,\theta)$ be of Hermitian symmetric type and let the notation be as above. Then for any $v\in V$, $\nu(v)\in W^{J}$ is the unique element in $Y_{v_{0}}(v)$.

Proof.

Note that $P=BK$. Let $v\in V$. By the definition of $\nu(v)$, one has

$$BvK\subset B\nu(v)P\subset\overline{B\nu(v)P}=\overline{B\nu(v)BK}=\overline{B\nu(v)B}\;K=\overline{B(m(\nu(v))\!\cdot\!v_{0})K}.$$

Thus $\nu(v)\in W_{v_{0}}(v)$. Conversely, assume that $w\in W_{v_{0}}(v)$. Write $w=dx$, where $d\in W^{J}$ and $x\in W_{J}$. Since every $\alpha\in J$ is compact imaginary for $v_{0}$, $m(x)\!\cdot\!v_{0}=v_{0}$. Thus $v\leq m(w)\!\cdot\!v_{0}=m(d)\!\cdot\!v_{0}$, i.e.,

$$BvK\subset\overline{B(m(d)\!\cdot\!v_{0})K}=\overline{BdB}\;K=\overline{BdP}.$$

Since $BvK\subset B\nu(v)P$, we have $B\nu(v)P\cap\overline{BdP}\neq\emptyset$. Thus $\nu(v)\leq d\leq w$. This shows that $\nu(v)$ is the unique minimal element in $W_{v_{0}}(v)$.

Q.E.D.

Thus, our Theorem 2.2 generalizes [13, Theorem 5.2.5].

9.4. An example where $Z_{v_{0}}(v)\neq Y_{v_{0}}(v)$

Let $G=SL(4,{\mathbb{C}})$ and let $\theta\in{\rm Aut}(G)$ be given by $\theta(g)=I_{2,2}(g^{t})^{-1}I_{2,2}$, where for $g\in G$, $g^{t}$ denotes the transpose of $g$, and $I_{2,2}={\rm diag}(I_{2},-I_{2})$ with $I_{2}$ being the $2\times 2$ identity matrix. Then $K=S(GL(2,{\mathbb{C}})\times GL(2,{\mathbb{C}}))$. Using the “kgb” command in the Atlas of Lie groups (www.liegroups.org) for the real form $SU(2,2)$ of $G$, one knows that there are $6$ elements in $V_{0}$. Take $v_{0}\in V_{0}$ to be the orbit labeled by $3$ in the Atlas and let $v$ be the orbit labeled by $19$. With respect to $v_{0}$, the simple roots $\alpha_{1}$ and $\alpha_{3}$ are noncompact, while $\alpha_{2}$ is compact, where we use the Bourbaki labeling of roots. Then $v=m(s_{3}s_{2}s_{1})\!\cdot\!v_{0}\in M(W,S)\!\cdot\!v_{0}$. By Example 8.11, $l(z)=3$ for every $z\in Z_{v_{0}}(v)$.

On the other hand, let $y=s_{2}s_{1}s_{2}s_{3}\in W$. Then $m(y)\!\cdot\!v_{0}=v_{{\rm max}}$, where $v_{{\rm max}}$ is maximal element in $V$. Since $v\leq v_{{\rm max}}$, $y\in W_{v_{0}}(v)$. We claim that $y\in Y_{v_{0}}(v)$. Indeed, if $y\notin Y_{v_{0}}(v)$, then there exists $y^{\prime}\in Y_{v_{0}}(v)$ with $y^{\prime}<y$. Since $l(y)\geq l(v)=3$, we must have $l(y)=3$. Now there are exactly three subwords $y^{\prime}$ of the word $y=s_{2}s_{1}s_{2}s_{3}$ with length $3$, namely, $y^{\prime}=s_{1}s_{2}s_{3}$ or $s_{2}s_{1}s_{3}$, or $s_{2}s_{1}s_{2}$, and one checks directly that neither of these three choices gives $v\leq m(y^{\prime})\!\cdot\!v_{0}$. Thus $y\in Y_{v_{0}}(v)$. Since $l(y)=4$, $y\notin Z_{v_{0}}(v)$. We thus have an example in which $Z_{v_{0}}(v)$ is a proper subset of $Y_{v_{0}}(v)$.